GCSE Maths: Roots and fractional indices (Higher)
Part of the GCSE Maths standard form, indices and surds chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Roots and fractional indices
A root undoes a power. A square has area 25 cm2. Its side is the number that multiplies by itself to give 25. That number is 5, so the square root of 25 is 5. We write √25 = 5. A cube has volume 27 cm3. Its edge is the number that gives 27 when multiplied by itself three times, so the cube root of 27 is 3, written 3√27 = 3.
Learn the perfect squares up to 152 = 225 and the cubes 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 (see Chapter 2). Then roots of these are quick. You can also take roots of fractions and decimals: √(9/16) = 3/4 because (3/4)2 = 9/16, and √0.25 = 0.5 because 0.5 × 0.5 = 0.25.
Most numbers are not perfect squares, but we can say where their roots lie. √50 is between √49 = 7 and √64 = 8, so it is between 7 and 8. A calculator gives √50 = 7.07…. Use the √ key for square roots and the cube root key (often SHIFT then √) for cube roots.
Remember: (√a)2 = a. Equations such as x2 = 49 have two solutions, x = 7 and x = −7, because (−7)2 = 49 too. The √ sign itself means the positive root. You cannot take the square root of a negative number (no real number squares to −9), but a cube root of a negative number is fine: 3√(−8) = −2.
Key ideas
- √a is the positive number that multiplies by itself to give a. 3√a multiplies three times.
- (√a)2 = a, and x2 = k has two solutions, ±√k.
- To place a root between two whole numbers use the perfect squares (or cubes) on each side.
- Roots of fractions: take the root of the top and of the bottom.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 2.
- Say what the question is asking. The square root of a number is the number that gives the original number when 2 lots of it are multiplied together. It undoes the power 2.√144: which number to the power 2 equals 144?
- Test the powers in order. List the powers until you reach the number.12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36, 72 = 49, 82 = 64, 92 = 81, 102 = 100, 112 = 121, 122 = 144
- Write the answer. 122 = 144.12
Check. Check: 122 = 12 × 12 = 144. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 2.
- Spot that the number is a fraction. A fraction is a top number over a bottom number, and we can deal with each separately.916: top 9, bottom 16
- Take the root of the top and of the bottom. For a fraction, √(a ÷ b) = √a ÷ √b.√9 = 3 and √16 = 4
- Write the answer as a fraction. 34
Check. Check: 34 to the power 2 = 916. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 3.
- Say what the question is asking. The cube root of a number is the number that gives the original number when 3 lots of it are multiplied together. It undoes the power 3.3√125: which number to the power 3 equals 125?
- Test the powers in order. List the powers until you reach the number.13 = 1, 23 = 8, 33 = 27, 43 = 64, 53 = 125
- Write the answer. 53 = 125.5
Check. Check: 53 = 5 × 5 × 5 = 125. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. A perfect square is the square of a whole number (1, 4, 9, 16…). A perfect cube is the cube of a whole number (1, 8, 27, 64…).
Method. Find the two perfect squares that the number lies between, then take roots.
- Find the perfect powers on either side of the number. We know the square root of perfect squares exactly, so use them as markers.72 = 49 and 82 = 64
- Place the number between them. 50 lies between 49 and 64.49 < 50 < 64
- Take the root of each part. Roots keep the order (a bigger number has a bigger root).7 < √50 < 8
Check. A calculator gives √50 = 7.071, which is between 7 and 8. ✓
Watch out. Between two whole numbers means one is below and the next one is above.
Method. Work out each root or power separately, then add and subtract.
- Work out each root or power on its own. A root undoes a power; a power is repeated multiplication.√36 = 6, because 6 × 6 = 36
3√27 = 3, because 3 × 3 × 3 = 27
23 = 2 × 2 × 2 = 8 - Put the values into the calculation. Keep the signs the same.√36 + 3√27 − 23 = 6 + 3 − 8
- Work out the answer. Add and subtract from left to right.6 + 3 − 8 = 1
Check. Work through it again from the left: 6 + 3 − 8 = 1. ✓
Watch out. Do each root or power first (BIDMAS: powers and roots before adding and subtracting).
Watch out
The square root of 36 is 6, not 18. A root does not halve the number. And √(9 + 16) is not √9 + √16: √25 = 5 but 3 + 4 = 7. Add or subtract inside the root first.Higher tier: fractional indices
The laws of indices tell us what a fractional index must mean. Take 251/2 and multiply it by itself: 251/2 × 251/2 = 251/2+1/2 = 251 = 25. So 251/2 is a number that multiplies by itself to give 25. That is the square root: 251/2 = √25 = 5. In the same way a1/3 is the cube root of a, and a1/n is the nth root of a.
For a general fraction, the denominator is the root and the numerator is the power: am/n = (n√a)m. For example 82/3 = (3√8)2 = 22 = 4. Take the root first: it keeps the numbers small. A negative fractional index combines both ideas: take the reciprocal, then the root and power. 16−3/4 = (1/16)3/4 = (1/2)3 = 1/8.
The same laws work with letters: (16x8)3/4 = 163/4 × x8×3/4 = 8x6. And you can solve equations: to undo a power p/q raise both sides to the power q/p. If the bases differ, write both as powers of the same number: 4x = 8 becomes (22)x = 23, so 2x = 3 and x = 3/2.
Key ideas (Higher)
- a1/2 = √a, a1/3 = 3√a, a1/n = nth root of a.
- am/n = (n√a)m: denominator is the root, numerator is the power. Do the root first.
- a−m/n = 1 ÷ am/n: flip first.
- To solve xp/q = k, raise both sides to the power q/p.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.82/3: denominator 3 means the cube root; numerator 2 means the power 2
- Take the cube root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 3 times gives the base?cube root of 8: 23 = 8, so it is 2.
- Raise the answer to the power 2. The numerator is the power.22 = 2 × 2 = 4
Check. Check with the other order: raise to the power first, then take the root. 82 = 64 and its cube root is 4. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.16−3/4: denominator 4 means the fourth root; numerator 3 means the power 3; the minus sign means the reciprocal
- Deal with the minus sign: take the reciprocal first. A negative index means "one over" (or flip the fraction). Doing this first lets us finish with a positive index.16−3/4 = (116)3/4
- Take the fourth root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 4 times gives the base?fourth root of 116: the fourth root of the top and of the bottom: 1 and 2, so it is 12.
- Raise the answer to the power 3. The numerator is the power.(12)3 = 18 = 18
Check. Check with the other order: raise to the power first, then take the root. 1163 = 14096 and its fourth root is 18. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.(278)−2/3: denominator 3 means the cube root; numerator 2 means the power 2; the minus sign means the reciprocal
- Deal with the minus sign: take the reciprocal first. A negative index means "one over" (or flip the fraction). Doing this first lets us finish with a positive index.(278)−2/3 = (827)2/3
- Take the cube root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 3 times gives the base?cube root of 827: the cube root of the top and of the bottom: 2 and 3, so it is 23.
- Raise the answer to the power 2. The numerator is the power.(23)2 = 49 = 49
Check. Check with the other order: raise to the power first, then take the root. 8272 = 64729 and its cube root is 49. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x3 the coefficient is 7. A letter on its own, like x, really means x1.
Method. Raise the number and every letter to the power. Multiply the indices: (am)n = am×n.
- Raise the number in front to the power. Everything inside the bracket is raised to the power, and that includes the number.The 4th root of 16 is 2, because 24 = 16.
Then 23 = 8 - Raise the letter x to the power: multiply the indices. (am)n = am×n.(x8)3/4 = x8 × 3/4 = x6
- Put it all together. 8x6
Check. Choose numbers: x = 4. The original expression has the value 32,768, and the answer 8x6 also has the value 32,768. ✓
Watch out. The number in front is raised to the power too: (2a)3 = 8a3, not 2a3.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Undo a fractional power by raising both sides to the reciprocal power.
- Spot the fractional index. The letter is raised to the power p/q. To get the letter on its own we need to undo that power.x3/2 = 27
- Raise both sides to the reciprocal power. The reciprocal of 3/2 is 2/3. Raising to this power gives (x3/2)2/3 = x1, which is just x. Do the same to the right-hand side to keep the equation balanced.x = 272/3
- Evaluate: root first, then power. The denominator 3 is the root and the numerator 2 is the power.the cube root of 27 is 3, then 32 = 9
x = 9
Check. Put x = 9 back: the square root of 9 is 3, and 33 = 27. ✓
Watch out. Use the reciprocal of the index (flip the fraction), not the negative of it.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Write both sides as powers of the same base, then compare the indices.
- Write both sides as powers of the same base. The bases 4 and 8 look different, but both are powers of 2.4 = 22
8 = 23 - Replace them in the equation. Use the power-of-a-power law: (am)n = am×n. So raising 22 to the power x multiplies the indices.(22)x = 22x, so 22x = 23
- Equate the indices. The bases are the same, so the indices must be equal.2x = 3
- Solve. Divide both sides by 2.x = 32 = 32
Check. Put x = 32 back: 43/2 = (22)3/2 = 22 × (3/2) = 23 = 8. ✓
Watch out. Find the common base first (2, 3 or 5 are the usual ones). Then the powers-of-a-power law turns the problem into a simple equation.
Watch out
a1/2 is the square root of a. It is not a ÷ 2. And a2/3 needs both the cube root and the square: forgetting the numerator is a very common error.Exercise 6E Roots without a calculator
Non-calculatorKey words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 2.
- Say what the question is asking. The square root of a number is the number that gives the original number when 2 lots of it are multiplied together. It undoes the power 2.√36: which number to the power 2 equals 36?
- Test the powers in order. List the powers until you reach the number.12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36
- Write the answer. 62 = 36.6
Check. Check: 62 = 6 × 6 = 36. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 2.
- Say what the question is asking. The square root of a number is the number that gives the original number when 2 lots of it are multiplied together. It undoes the power 2.√81: which number to the power 2 equals 81?
- Test the powers in order. List the powers until you reach the number.12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36, 72 = 49, 82 = 64, 92 = 81
- Write the answer. 92 = 81.9
Check. Check: 92 = 9 × 9 = 81. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 2.
- Say what the question is asking. The square root of a number is the number that gives the original number when 2 lots of it are multiplied together. It undoes the power 2.√121: which number to the power 2 equals 121?
- Test the powers in order. List the powers until you reach the number.12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36, 72 = 49, 82 = 64, 92 = 81, 102 = 100, 112 = 121
- Write the answer. 112 = 121.11
Check. Check: 112 = 11 × 11 = 121. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 2.
- Say what the question is asking. The square root of a number is the number that gives the original number when 2 lots of it are multiplied together. It undoes the power 2.√169: which number to the power 2 equals 169?
- Test the powers in order. List the powers until you reach the number.12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36, 72 = 49, 82 = 64, 92 = 81, 102 = 100, 112 = 121, 122 = 144, 132 = 169
- Write the answer. 132 = 169.13
Check. Check: 132 = 13 × 13 = 169. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 2.
- Say what the question is asking. The square root of a number is the number that gives the original number when 2 lots of it are multiplied together. It undoes the power 2.√400: which number to the power 2 equals 400?
- Test the powers in order. List the powers until you reach the number.12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36, 72 = 49, 82 = 64, 92 = 81, 102 = 100, 112 = 121, 122 = 144, 132 = 169, 142 = 196, 152 = 225, 162 = 256, 172 = 289, 182 = 324, 192 = 361, 202 = 400
- Write the answer. 202 = 400.20
Check. Check: 202 = 20 × 20 = 400. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 2.
- Say what the question is asking. The square root of a number is the number that gives the original number when 2 lots of it are multiplied together. It undoes the power 2.√25: which number to the power 2 equals 25?
- Test the powers in order. List the powers until you reach the number.12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25
- Write the answer. 52 = 25.5
Check. Check: 52 = 5 × 5 = 25. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 3.
- Say what the question is asking. The cube root of a number is the number that gives the original number when 3 lots of it are multiplied together. It undoes the power 3.3√8: which number to the power 3 equals 8?
- Test the powers in order. List the powers until you reach the number.13 = 1, 23 = 8
- Write the answer. 23 = 8.2
Check. Check: 23 = 2 × 2 × 2 = 8. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 3.
- Say what the question is asking. The cube root of a number is the number that gives the original number when 3 lots of it are multiplied together. It undoes the power 3.3√27: which number to the power 3 equals 27?
- Test the powers in order. List the powers until you reach the number.13 = 1, 23 = 8, 33 = 27
- Write the answer. 33 = 27.3
Check. Check: 33 = 3 × 3 × 3 = 27. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 3.
- Say what the question is asking. The cube root of a number is the number that gives the original number when 3 lots of it are multiplied together. It undoes the power 3.3√64: which number to the power 3 equals 64?
- Test the powers in order. List the powers until you reach the number.13 = 1, 23 = 8, 33 = 27, 43 = 64
- Write the answer. 43 = 64.4
Check. Check: 43 = 4 × 4 × 4 = 64. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 3.
- Say what the question is asking. The cube root of a number is the number that gives the original number when 3 lots of it are multiplied together. It undoes the power 3.3√125: which number to the power 3 equals 125?
- Test the powers in order. List the powers until you reach the number.13 = 1, 23 = 8, 33 = 27, 43 = 64, 53 = 125
- Write the answer. 53 = 125.5
Check. Check: 53 = 5 × 5 × 5 = 125. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 3.
- Say what the question is asking. The cube root of a number is the number that gives the original number when 3 lots of it are multiplied together. It undoes the power 3.3√1000: which number to the power 3 equals 1000?
- Test the powers in order. List the powers until you reach the number.13 = 1, 23 = 8, 33 = 27, 43 = 64, 53 = 125, 63 = 216, 73 = 343, 83 = 512, 93 = 729, 103 = 1000
- Write the answer. 103 = 1000.10
Check. Check: 103 = 10 × 10 × 10 = 1000. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 3.
- Say what the question is asking. The cube root of a number is the number that gives the original number when 3 lots of it are multiplied together. It undoes the power 3.3√216: which number to the power 3 equals 216?
- Test the powers in order. List the powers until you reach the number.13 = 1, 23 = 8, 33 = 27, 43 = 64, 53 = 125, 63 = 216
- Write the answer. 63 = 216.6
Check. Check: 63 = 6 × 6 × 6 = 216. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 2.
- Spot that the number is a fraction. A fraction is a top number over a bottom number, and we can deal with each separately.916: top 9, bottom 16
- Take the root of the top and of the bottom. For a fraction, √(a ÷ b) = √a ÷ √b.√9 = 3 and √16 = 4
- Write the answer as a fraction. 34
Check. Check: 34 to the power 2 = 916. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 2.
- Spot that the number is a fraction. A fraction is a top number over a bottom number, and we can deal with each separately.425: top 4, bottom 25
- Take the root of the top and of the bottom. For a fraction, √(a ÷ b) = √a ÷ √b.√4 = 2 and √25 = 5
- Write the answer as a fraction. 25
Check. Check: 25 to the power 2 = 425. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 2.
- Spot that the number is a fraction. A fraction is a top number over a bottom number, and we can deal with each separately.136: top 1, bottom 36
- Take the root of the top and of the bottom. For a fraction, √(a ÷ b) = √a ÷ √b.√1 = 1 and √36 = 6
- Write the answer as a fraction. 16
Check. Check: 16 to the power 2 = 136. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 2.
- Spot that the number is a fraction. A fraction is a top number over a bottom number, and we can deal with each separately.4964: top 49, bottom 64
- Take the root of the top and of the bottom. For a fraction, √(a ÷ b) = √a ÷ √b.√49 = 7 and √64 = 8
- Write the answer as a fraction. 78
Check. Check: 78 to the power 2 = 4964. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 2.
- Write the decimal as a fraction. A fraction lets us take the root of the top and the bottom separately.0.25 = 25100
- Take the square root of the top and of the bottom. √(a ÷ b) = √a ÷ √b.√25 = 5 and √100 = 10, so 510
- Write the answer as a decimal. Convert the fraction back.0.5
Check. Check: 0.5 × 0.5 = 0.25. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 2.
- Write the decimal as a fraction. A fraction lets us take the root of the top and the bottom separately.0.81 = 81100
- Take the square root of the top and of the bottom. √(a ÷ b) = √a ÷ √b.√81 = 9 and √100 = 10, so 910
- Write the answer as a decimal. Convert the fraction back.0.9
Check. Check: 0.9 × 0.9 = 0.81. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 2.
- Write the decimal as a fraction. A fraction lets us take the root of the top and the bottom separately.0.0144 = 14410,000
- Take the square root of the top and of the bottom. √(a ÷ b) = √a ÷ √b.√144 = 12 and √10,000 = 100, so 12100
- Write the answer as a decimal. Convert the fraction back.0.12
Check. Check: 0.12 × 0.12 = 0.0144. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 3.
- Write the decimal as a fraction. A fraction lets us take the root of the top and the bottom separately.0.027 = 271000
- Take the cube root of the top and of the bottom. 3√(a ÷ b) = 3√a ÷ 3√b.3√27 = 3 and 3√1000 = 10, so 310
- Write the answer as a decimal. Convert the fraction back.0.3
Check. Check: 0.3 × 0.3 × 0.3 = 0.027. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Key words. The square root of a number undoes squaring: √49 = 7 because 72 = 49. The cube root undoes cubing: 3√27 = 3 because 33 = 27.
Method. Find the number that gives the original when raised to the power 3.
- Write the decimal as a fraction. A fraction lets us take the root of the top and the bottom separately.0.008 = 81000
- Take the cube root of the top and of the bottom. 3√(a ÷ b) = 3√a ÷ 3√b.3√8 = 2 and 3√1000 = 10, so 210
- Write the answer as a decimal. Convert the fraction back.0.2
Check. Check: 0.2 × 0.2 × 0.2 = 0.008. ✓
Watch out. The square root of 49 is 7, not 24.5. Do not halve the number: a root undoes a power.
Method. Work out each root or power separately, then add and subtract.
- Work out each root or power on its own. A root undoes a power; a power is repeated multiplication.√49 = 7, because 7 × 7 = 49
√81 = 9, because 9 × 9 = 81 - Put the values into the calculation. Keep the signs the same.√49 + √81 = 7 + 9
- Work out the answer. Add and subtract from left to right.7 + 9 = 16
Check. Work through it again from the left: 7 + 9 = 16. ✓
Watch out. Do each root or power first (BIDMAS: powers and roots before adding and subtracting).
Method. Work out each root or power separately, then add and subtract.
- Work out each root or power on its own. A root undoes a power; a power is repeated multiplication.3√64 = 4, because 4 × 4 × 4 = 64
√16 = 4, because 4 × 4 = 16 - Put the values into the calculation. Keep the signs the same.3√64 − √16 = 4 − 4
- Work out the answer. Add and subtract from left to right.4 − 4 = 0
Check. Work through it again from the left: 4 − 4 = 0. ✓
Watch out. Do each root or power first (BIDMAS: powers and roots before adding and subtracting).
Method. Work out each root or power separately, then add and subtract.
- Work out each root or power on its own. A root undoes a power; a power is repeated multiplication.52 = 5 × 5 = 25
3√125 = 5, because 5 × 5 × 5 = 125 - Put the values into the calculation. Keep the signs the same.52 − 3√125 = 25 − 5
- Work out the answer. Add and subtract from left to right.25 − 5 = 20
Check. Work through it again from the left: 25 − 5 = 20. ✓
Watch out. Do each root or power first (BIDMAS: powers and roots before adding and subtracting).
Method. Work out each root or power separately, then add and subtract.
- Work out each root or power on its own. A root undoes a power; a power is repeated multiplication.√100 = 10, because 10 × 10 = 100
33 = 3 × 3 × 3 = 27
√4 = 2, because 2 × 2 = 4 - Put the values into the calculation. Keep the signs the same.√100 + 33 − √4 = 10 + 27 − 2
- Work out the answer. Add and subtract from left to right.10 + 27 − 2 = 35
Check. Work through it again from the left: 10 + 27 − 2 = 35. ✓
Watch out. Do each root or power first (BIDMAS: powers and roots before adding and subtracting).
Key words. A perfect square is the square of a whole number (1, 4, 9, 16…). A perfect cube is the cube of a whole number (1, 8, 27, 64…).
Method. Find the two perfect squares that the number lies between, then take roots.
- Find the perfect powers on either side of the number. We know the square root of perfect squares exactly, so use them as markers.42 = 16 and 52 = 25
- Place the number between them. 20 lies between 16 and 25.16 < 20 < 25
- Take the root of each part. Roots keep the order (a bigger number has a bigger root).4 < √20 < 5
Check. A calculator gives √20 = 4.472, which is between 4 and 5. ✓
Watch out. Between two whole numbers means one is below and the next one is above.
Key words. A perfect square is the square of a whole number (1, 4, 9, 16…). A perfect cube is the cube of a whole number (1, 8, 27, 64…).
Method. Find the two perfect squares that the number lies between, then take roots.
- Find the perfect powers on either side of the number. We know the square root of perfect squares exactly, so use them as markers.82 = 64 and 92 = 81
- Place the number between them. 70 lies between 64 and 81.64 < 70 < 81
- Take the root of each part. Roots keep the order (a bigger number has a bigger root).8 < √70 < 9
Check. A calculator gives √70 = 8.367, which is between 8 and 9. ✓
Watch out. Between two whole numbers means one is below and the next one is above.
Key words. A perfect square is the square of a whole number (1, 4, 9, 16…). A perfect cube is the cube of a whole number (1, 8, 27, 64…).
Method. Find the two perfect cubes that the number lies between, then take roots.
- Find the perfect powers on either side of the number. We know the cube root of perfect cubes exactly, so use them as markers.33 = 27 and 43 = 64
- Place the number between them. 30 lies between 27 and 64.27 < 30 < 64
- Take the root of each part. Roots keep the order (a bigger number has a bigger root).3 < 3√30 < 4
Check. A calculator gives 3√30 = 3.107, which is between 3 and 4. ✓
Watch out. Between two whole numbers means one is below and the next one is above.
Key words. A perfect square is the square of a whole number (1, 4, 9, 16…). A perfect cube is the cube of a whole number (1, 8, 27, 64…).
Method. Find the two perfect cubes that the number lies between, then take roots.
- Find the perfect powers on either side of the number. We know the cube root of perfect cubes exactly, so use them as markers.43 = 64 and 53 = 125
- Place the number between them. 100 lies between 64 and 125.64 < 100 < 125
- Take the root of each part. Roots keep the order (a bigger number has a bigger root).4 < 3√100 < 5
Check. A calculator gives 3√100 = 4.642, which is between 4 and 5. ✓
Watch out. Between two whole numbers means one is below and the next one is above.
Key words. A perfect square is the square of a whole number (1, 4, 9, 16…). A perfect cube is the cube of a whole number (1, 8, 27, 64…).
Method. Find the two perfect squares that the number lies between, then take roots.
- Find the perfect powers on either side of the number. We know the square root of perfect squares exactly, so use them as markers.142 = 196 and 152 = 225
- Place the number between them. 200 lies between 196 and 225.196 < 200 < 225
- Take the root of each part. Roots keep the order (a bigger number has a bigger root).14 < √200 < 15
Check. A calculator gives √200 = 14.142, which is between 14 and 15. ✓
Watch out. Between two whole numbers means one is below and the next one is above.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Take the square root of both sides. There are two answers: one positive and one negative.
- Say what the equation asks. We need a number which, multiplied by itself 2 times, makes 49.x2 = 49
- Find the positive root. 7 × 7 = 49.72 = 49
- Remember the negative root. A negative number squared is also positive: (−7)2 = 49. An even power cannot tell the sign.x = 7 or x = −7
Check. 72 = 49 and (−7)2 = 49. ✓
Watch out. Do not forget the negative answer when you solve an equation with x2.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Take the square root of both sides. There are two answers: one positive and one negative.
- Say what the equation asks. We need a number which, multiplied by itself 2 times, makes 121.x2 = 121
- Find the positive root. 11 × 11 = 121.112 = 121
- Remember the negative root. A negative number squared is also positive: (−11)2 = 121. An even power cannot tell the sign.x = 11 or x = −11
Check. 112 = 121 and (−11)2 = 121. ✓
Watch out. Do not forget the negative answer when you solve an equation with x2.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Take the cube root of both sides. A cube root gives just one answer.
- Say what the equation asks. We need a number which, cubed, makes 27.x3 = 27
- Find the cube root. 3 × 3 × 3 = 27.x = 3√27 = 3
- Check for other answers. A cube of a negative number is negative, so there is only one answer.x = 3
Check. 33 = 27. ✓
Watch out. Odd powers have only one real solution.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Take the cube root of both sides. A cube root gives just one answer.
- Say what the equation asks. We need a number which, cubed, makes 125.x3 = 125
- Find the cube root. 5 × 5 × 5 = 125.x = 3√125 = 5
- Check for other answers. A cube of a negative number is negative, so there is only one answer.x = 5
Check. 53 = 125. ✓
Watch out. Odd powers have only one real solution.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Squaring and square-rooting are inverse operations, so they undo each other.
- Remember what a square root is. √7 is the number which, multiplied by itself, makes 7.√7 × √7 = 7
- Squaring undoes the square root. Squaring means multiplying by itself.(√7)2 = √7 × √7 = 7
Check. A calculator gives √7 = 2.6458… and its square is 7. ✓
Watch out. (√7)2 is 7, not 49 and not 3.5.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Squaring and square-rooting are inverse operations, so they undo each other.
- Remember what a square root is. √13 is the number which, multiplied by itself, makes 13.√13 × √13 = 13
- Squaring undoes the square root. Squaring means multiplying by itself.(√13)2 = √13 × √13 = 13
Check. A calculator gives √13 = 3.6056… and its square is 13. ✓
Watch out. (√7)2 is 7, not 49 and not 3.5.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Squaring and square-rooting are inverse operations, so they undo each other.
- Remember what a square root is. √2 is the number which, multiplied by itself, makes 2.√2 × √2 = 2
- Squaring undoes the square root. Squaring means multiplying by itself.(√2)2 = √2 × √2 = 2
Check. A calculator gives √2 = 1.4142… and its square is 2. ✓
Watch out. (√7)2 is 7, not 49 and not 3.5.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Squaring and square-rooting are inverse operations, so they undo each other.
- Remember what a square root is. √50 is the number which, multiplied by itself, makes 50.√50 × √50 = 50
- Squaring undoes the square root. Squaring means multiplying by itself.(√50)2 = √50 × √50 = 50
Check. A calculator gives √50 = 7.0711… and its square is 50. ✓
Watch out. (√7)2 is 7, not 49 and not 3.5.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. The side of a square is the square root of its area. The perimeter is 4 times the side.
- Find the side. Side × side = area, so side = √area.√81 = 9, because 9 × 9 = 81. The side is 9 m.
- Find the perimeter. A square has 4 equal sides.4 × 9 = 36
- Write the answers. Side 9 m; fence 36 m.
Check. 9 × 9 = 81 and 9 + 9 + 9 + 9 = 36. ✓
Watch out. Perimeter is a length (m); area is in m2.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Find the edge of the cube using a cube root, then multiply by 12.
- Find the length of one edge. Edge × edge × edge = volume, so edge = 3√volume.3√216 = 6, because 6 × 6 × 6 = 216
- Multiply by 12. A cube has 12 equal edges.12 × 6 = 72
- Write the answer. 72 cm
Check. 6 × 6 = 36, 36 × 6 = 216. ✓
Watch out. Use a cube root for a volume of a cube, not a square root.
Key words. A perfect square is the square of a whole number (1, 4, 9, 16…). A perfect cube is the cube of a whole number (1, 8, 27, 64…).
Method. Find the two perfect squares that the number lies between, then take roots.
- Find the perfect powers on either side of the number. We know the square root of perfect squares exactly, so use them as markers.122 = 144 and 132 = 169
- Place the number between them. 150 lies between 144 and 169.144 < 150 < 169
- Take the root of each part. Roots keep the order (a bigger number has a bigger root).12 < √150 < 13
Check. A calculator gives √150 = 12.247, which is between 12 and 13. ✓
Watch out. Between two whole numbers means one is below and the next one is above.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Work out both and compare.
- Work out the left side. Add inside the root first.√(9 + 16) = √25 = 5
- Work out the right side. Take each root separately and then add.√9 + √16 = 3 + 4 = 7
- Compare. 5 is not equal to 7.No, they are not the same.
Check. 5 × 5 = 25 and 7 × 7 = 49, so the two values clearly differ. ✓
Watch out. A root of a sum is not the sum of the roots.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Square the two numbers and compare with 50. Squaring keeps the order for positive numbers.
- Square 7. 72 = 49
- Square 7.1. Multiply 71 × 71 = 5041, then put in two decimal places.7.12 = 50.41
- Compare. Since 49 < 50 < 50.41, taking square roots of each part keeps the order.7 < √50 < 7.1
Check. A calculator gives √50 = 7.0710…. ✓
Watch out. Write the squared values next to 50 to make the comparison clear.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Find a number which, cubed, gives −8.
- Think about the sign. A negative number multiplied by itself three times is negative, so the answer is negative.(−2) × (−2) = 4 and 4 × (−2) = −8
- Write the answer. 3√(−8) = −2
Check. (−2)3 = −8. ✓
Watch out. You can take the cube root of a negative number but not the square root.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Find a number which, cubed, gives −27.
- Think about the sign. The answer must be negative.(−3) × (−3) = 9 and 9 × (−3) = −27
- Write the answer. 3√(−27) = −3
Check. (−3)3 = −27. ✓
Watch out. Check the sign by cubing your answer.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Think about the sign of a square.
- Squares of positive numbers. 3 × 3 = 9 is positive.32 = 9
- Squares of negative numbers. (−3) × (−3) = 9, because negative × negative = positive.(−3)2 = 9
- Conclude. Every square is zero or positive, so none can equal −9.No real number squares to −9.
Check. Try 0: 02 = 0, which is not negative either. ✓
Watch out. Be careful to separate √(−9) from −√9 = −3, which is fine.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. List the squares and cubes in the range and look for a number in both lists.
- List the squares between 50 and 100. Square the whole numbers from 8 to 10.82 = 64, 92 = 81, 102 = 100 (72 = 49 is too small)
- List the cubes between 50 and 100. Cube 4 and 5.43 = 64, 53 = 125 (too big)
- Find the number in both lists. 64 (= 82 = 43)
Check. 8 × 8 = 64 and 4 × 4 × 4 = 64. ✓
Watch out. Only 64 works in this range (also 26).
Exercise 6F Powers and roots with a calculator
Calculator allowedMethod. Use the root key on the calculator and round to the accuracy asked for.
- Enter the calculation. Press the root key first, then the number (in a bracket if your calculator needs one).Press: [√] 7 [=]
- Read the display. The calculator shows many digits.√7 = 2.645751…
- Round to 3 significant figures. Count 3 digits from the first non-zero digit, then look at the next digit to decide whether to round up.2.65
Check. The answer should lie between 2 and 3, because 22 = 4 and 32 = 9. ✓
Watch out. Round only at the end, and give the number of significant figures asked for.
Method. Use the root key on the calculator and round to the accuracy asked for.
- Enter the calculation. Press the root key first, then the number (in a bracket if your calculator needs one).Press: [√] 130 [=]
- Read the display. The calculator shows many digits.√130 = 11.401754…
- Round to 3 significant figures. Count 3 digits from the first non-zero digit, then look at the next digit to decide whether to round up.11.4
Check. The answer should lie between 11 and 12, because 112 = 121 and 122 = 144. ✓
Watch out. Round only at the end, and give the number of significant figures asked for.
Method. Use the root key on the calculator and round to the accuracy asked for.
- Enter the calculation. Press the root key first, then the number (in a bracket if your calculator needs one).Press: [√] 2000 [=]
- Read the display. The calculator shows many digits.√2000 = 44.72136…
- Round to 3 significant figures. Count 3 digits from the first non-zero digit, then look at the next digit to decide whether to round up.44.7
Check. The answer should lie between 44 and 45, because 442 = 1936 and 452 = 2025. ✓
Watch out. Round only at the end, and give the number of significant figures asked for.
Method. Use the root key on the calculator and round to the accuracy asked for.
- Enter the calculation. Press the root key first, then the number (in a bracket if your calculator needs one).Press: [SHIFT] [√] (the cube root key, shown as 3√) 50 [=]
- Read the display. The calculator shows many digits.3√50 = 3.684031…
- Round to 3 significant figures. Count 3 digits from the first non-zero digit, then look at the next digit to decide whether to round up.3.68
Check. The answer should lie between 3 and 4, because 33 = 27 and 43 = 64. ✓
Watch out. Round only at the end, and give the number of significant figures asked for.
Method. Use the root key on the calculator and round to the accuracy asked for.
- Enter the calculation. Press the root key first, then the number (in a bracket if your calculator needs one).Press: [SHIFT] [√] (the cube root key, shown as 3√) 300 [=]
- Read the display. The calculator shows many digits.3√300 = 6.69433…
- Round to 3 significant figures. Count 3 digits from the first non-zero digit, then look at the next digit to decide whether to round up.6.69
Check. The answer should lie between 6 and 7, because 63 = 216 and 73 = 343. ✓
Watch out. Round only at the end, and give the number of significant figures asked for.
Method. Use the power key on the calculator (negative indices in brackets) and round at the end.
- Enter the calculation using the power key. The power key is labelled [x□] or [^]. Type the base, the key, then the index. For a negative index use the (−) key. Put a negative index in brackets.Press: 2.5 [^] 4 [=]
- Read the display. The calculator shows more digits than we need.2.54 = 39.0625…
- Round to 3 significant figures. Count 3 digits from the first non-zero digit, and look at the next digit to round.39.1
Check. Split the power: 2.52 × 2.52 = 6.25 × 6.25 = 39.0625. ✓
Watch out. Do not round in the middle of a calculation. For a negative index remember it means "one over the power".
Method. Use the power key on the calculator (negative indices in brackets) and round at the end.
- Enter the calculation using the power key. The power key is labelled [x□] or [^]. Type the base, the key, then the index. For a negative index use the (−) key. Put a negative index in brackets.Press: 1.05 [^] 10 [=]
- Read the display. The calculator shows more digits than we need.1.0510 = 1.628895…
- Round to 3 significant figures. Count 3 digits from the first non-zero digit, and look at the next digit to round.1.63
Check. Split the power: 1.055 × 1.055 = 1.2762815625 × 1.2762815625 = 1.62889462677744140625. ✓
Watch out. Do not round in the middle of a calculation. For a negative index remember it means "one over the power".
Method. Use the power key on the calculator (negative indices in brackets) and round at the end.
- Enter the calculation using the power key. The power key is labelled [x□] or [^]. Type the base, the key, then the index. For a negative index use the (−) key. Put a negative index in brackets.Press: 3.2 [^] (−2) [=]
- Read the display. The calculator shows more digits than we need.3.2−2 = 0.097656…
- Round to 3 significant figures. Count 3 digits from the first non-zero digit, and look at the next digit to round.0.0977
Check. Split the power: 3.21 × 3.21 = 3.2 × 3.2 = 10.24, and the reciprocal is 1 ÷ 10.24 = 0.097656…. ✓
Watch out. Do not round in the middle of a calculation. For a negative index remember it means "one over the power".
Method. Use the power key on the calculator (negative indices in brackets) and round at the end.
- Enter the calculation using the power key. The power key is labelled [x□] or [^]. Type the base, the key, then the index. For a negative index use the (−) key. Put a negative index in brackets.Press: 0.9 [^] 5 [=]
- Read the display. The calculator shows more digits than we need.0.95 = 0.59049…
- Round to 3 significant figures. Count 3 digits from the first non-zero digit, and look at the next digit to round.0.590
Check. Split the power: 0.92 × 0.93 = 0.81 × 0.729 = 0.59049. ✓
Watch out. Do not round in the middle of a calculation. For a negative index remember it means "one over the power".
Method. Use the power key on the calculator (negative indices in brackets) and round at the end.
- Enter the calculation using the power key. The power key is labelled [x□] or [^]. Type the base, the key, then the index. For a negative index use the (−) key. Put a negative index in brackets.Press: 8 [^] (−2) [=]
- Read the display. The calculator shows more digits than we need.8−2 = 0.015625…
- Round to 3 significant figures. Count 3 digits from the first non-zero digit, and look at the next digit to round.0.0156
Check. Split the power: 81 × 81 = 8 × 8 = 64, and the reciprocal is 1 ÷ 64 = 0.015625…. ✓
Watch out. Do not round in the middle of a calculation. For a negative index remember it means "one over the power".
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Side = √area. Use the root key and round to 1 decimal place.
- Write the side as a square root. Side × side = area.side = √90
- Use the calculator. Press [√] 90 [=].√90 = 9.4868…
- Round to 1 decimal place. The digit after the first decimal place is 8, so round up.9.5 cm
Check. 9.5 × 9.5 = 90.25, which is close to 90. ✓
Watch out. Give the unit and the accuracy asked for.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Work out the power first, then multiply. Round only at the very end.
- Work out 1.045. Use the power key: 1.04 [^] 5 [=].1.045 = 1.2166529024
- Multiply by 1500. Keep all the digits for now.1500 × 1.2166529024 = 1824.9793536
- Round to the nearest penny. Two decimal places.£1,824.98
Check. 1.04 × 1.04 × 1.04 × 1.04 × 1.04 ≈ 1.2167, and 1500 × 1.2167 ≈ 1825. ✓
Watch out. Do the power before multiplying (BIDMAS).
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Do the powers first, add, then take the square root (BIDMAS).
- Square each number. Use the x2 key or multiply by itself.5.22 = 27.04
3.92 = 15.21 - Add. The square root covers the whole sum.27.04 + 15.21 = 42.25
- Take the square root. Press [√] 42.25 [=].√42.25 = 6.5
Check. 6.5 × 6.5 = 42.25. ✓
Watch out. Add under the root before square-rooting.
Method. Use the root key on the calculator and round to the accuracy asked for.
- Enter the calculation. Press the root key first, then the number (in a bracket if your calculator needs one).Press: [√] 11 [=]
- Read the display. The calculator shows many digits.√11 = 3.316625…
- Round to 3 significant figures. Count 3 digits from the first non-zero digit, then look at the next digit to decide whether to round up.3.32
Check. The answer should lie between 3 and 4, because 32 = 9 and 42 = 16. ✓
Watch out. Round only at the end, and give the number of significant figures asked for.
Method. Use the root key on the calculator and round to the accuracy asked for.
- Enter the calculation. Press the root key first, then the number (in a bracket if your calculator needs one).Press: [√] 59 [=]
- Read the display. The calculator shows many digits.√59 = 7.681146…
- Round to 3 significant figures. Count 3 digits from the first non-zero digit, then look at the next digit to decide whether to round up.7.68
Check. The answer should lie between 7 and 8, because 72 = 49 and 82 = 64. ✓
Watch out. Round only at the end, and give the number of significant figures asked for.
Method. Use the power key on the calculator (negative indices in brackets) and round at the end.
- Enter the calculation using the power key. The power key is labelled [x□] or [^]. Type the base, the key, then the index. For a negative index use the (−) key. Put a negative index in brackets.Press: 1.5 [^] 6 [=]
- Read the display. The calculator shows more digits than we need.1.56 = 11.390625…
- Round to 3 significant figures. Count 3 digits from the first non-zero digit, and look at the next digit to round.11.4
Check. Split the power: 1.53 × 1.53 = 3.375 × 3.375 = 11.390625. ✓
Watch out. Do not round in the middle of a calculation. For a negative index remember it means "one over the power".
Method. Use the power key on the calculator (negative indices in brackets) and round at the end.
- Enter the calculation using the power key. The power key is labelled [x□] or [^]. Type the base, the key, then the index. For a negative index use the (−) key. Put a negative index in brackets.Press: 0.5 [^] (−3) [=]
- Read the display. The calculator shows more digits than we need.0.5−3 = 8…
- Round to 3 significant figures. Count 3 digits from the first non-zero digit, and look at the next digit to round.8.00
Check. Split the power: 0.51 × 0.52 = 0.5 × 0.25 = 0.125, and the reciprocal is 1 ÷ 0.125 = 8…. ✓
Watch out. Do not round in the middle of a calculation. For a negative index remember it means "one over the power".
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Cube both numbers and compare with 100.
- Cube 4.6. Use the power key: 4.6 [^] 3.4.63 = 97.336
- Cube 4.7. 4.73 = 103.823
- Compare. Cube roots keep the order, so 100 lies between the two cubes.97.336 < 100 < 103.823, so 4.6 < 3√100 < 4.7
Check. A calculator gives 3√100 = 4.6416…. ✓
Watch out. Show the two cubes in your answer.
Exercise 6G Fractional indices
Non-calculatorKey words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.251/2: denominator 2 means the square root; numerator 1 means the power 1
- Take the square root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself once more gives the base?square root of 25: 52 = 25, so it is 5.
- The numerator is 1, so there is no power to apply. The answer is just the root.5
Check. Check with the other order: raise to the power first, then take the root. 251 = 25 and its square root is 5. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.811/2: denominator 2 means the square root; numerator 1 means the power 1
- Take the square root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself once more gives the base?square root of 81: 92 = 81, so it is 9.
- The numerator is 1, so there is no power to apply. The answer is just the root.9
Check. Check with the other order: raise to the power first, then take the root. 811 = 81 and its square root is 9. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.641/3: denominator 3 means the cube root; numerator 1 means the power 1
- Take the cube root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 3 times gives the base?cube root of 64: 43 = 64, so it is 4.
- The numerator is 1, so there is no power to apply. The answer is just the root.4
Check. Check with the other order: raise to the power first, then take the root. 641 = 64 and its cube root is 4. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.1251/3: denominator 3 means the cube root; numerator 1 means the power 1
- Take the cube root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 3 times gives the base?cube root of 125: 53 = 125, so it is 5.
- The numerator is 1, so there is no power to apply. The answer is just the root.5
Check. Check with the other order: raise to the power first, then take the root. 1251 = 125 and its cube root is 5. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.161/4: denominator 4 means the fourth root; numerator 1 means the power 1
- Take the fourth root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 4 times gives the base?fourth root of 16: 24 = 16, so it is 2.
- The numerator is 1, so there is no power to apply. The answer is just the root.2
Check. Check with the other order: raise to the power first, then take the root. 161 = 16 and its fourth root is 2. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.321/5: denominator 5 means the fifth root; numerator 1 means the power 1
- Take the fifth root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 5 times gives the base?fifth root of 32: 25 = 32, so it is 2.
- The numerator is 1, so there is no power to apply. The answer is just the root.2
Check. Check with the other order: raise to the power first, then take the root. 321 = 32 and its fifth root is 2. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.82/3: denominator 3 means the cube root; numerator 2 means the power 2
- Take the cube root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 3 times gives the base?cube root of 8: 23 = 8, so it is 2.
- Raise the answer to the power 2. The numerator is the power.22 = 2 × 2 = 4
Check. Check with the other order: raise to the power first, then take the root. 82 = 64 and its cube root is 4. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.93/2: denominator 2 means the square root; numerator 3 means the power 3
- Take the square root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself once more gives the base?square root of 9: 32 = 9, so it is 3.
- Raise the answer to the power 3. The numerator is the power.33 = 3 × 3 × 3 = 27
Check. Check with the other order: raise to the power first, then take the root. 93 = 729 and its square root is 27. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.163/4: denominator 4 means the fourth root; numerator 3 means the power 3
- Take the fourth root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 4 times gives the base?fourth root of 16: 24 = 16, so it is 2.
- Raise the answer to the power 3. The numerator is the power.23 = 2 × 2 × 2 = 8
Check. Check with the other order: raise to the power first, then take the root. 163 = 4096 and its fourth root is 8. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.272/3: denominator 3 means the cube root; numerator 2 means the power 2
- Take the cube root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 3 times gives the base?cube root of 27: 33 = 27, so it is 3.
- Raise the answer to the power 2. The numerator is the power.32 = 3 × 3 = 9
Check. Check with the other order: raise to the power first, then take the root. 272 = 729 and its cube root is 9. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.45/2: denominator 2 means the square root; numerator 5 means the power 5
- Take the square root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself once more gives the base?square root of 4: 22 = 4, so it is 2.
- Raise the answer to the power 5. The numerator is the power.25 = 2 × 2 × 2 × 2 × 2 = 32
Check. Check with the other order: raise to the power first, then take the root. 45 = 1024 and its square root is 32. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.1252/3: denominator 3 means the cube root; numerator 2 means the power 2
- Take the cube root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 3 times gives the base?cube root of 125: 53 = 125, so it is 5.
- Raise the answer to the power 2. The numerator is the power.52 = 5 × 5 = 25
Check. Check with the other order: raise to the power first, then take the root. 1252 = 15,625 and its cube root is 25. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.4−1/2: denominator 2 means the square root; numerator 1 means the power 1; the minus sign means the reciprocal
- Deal with the minus sign: take the reciprocal first. A negative index means "one over" (or flip the fraction). Doing this first lets us finish with a positive index.4−1/2 = (14)1/2
- Take the square root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself once more gives the base?square root of 14: the square root of the top and of the bottom: 1 and 2, so it is 12.
- The numerator is 1, so there is no power to apply. The answer is just the root.12
Check. Check with the other order: raise to the power first, then take the root. 141 = 14 and its square root is 12. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.8−2/3: denominator 3 means the cube root; numerator 2 means the power 2; the minus sign means the reciprocal
- Deal with the minus sign: take the reciprocal first. A negative index means "one over" (or flip the fraction). Doing this first lets us finish with a positive index.8−2/3 = (18)2/3
- Take the cube root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 3 times gives the base?cube root of 18: the cube root of the top and of the bottom: 1 and 2, so it is 12.
- Raise the answer to the power 2. The numerator is the power.(12)2 = 14 = 14
Check. Check with the other order: raise to the power first, then take the root. 182 = 164 and its cube root is 14. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.27−1/3: denominator 3 means the cube root; numerator 1 means the power 1; the minus sign means the reciprocal
- Deal with the minus sign: take the reciprocal first. A negative index means "one over" (or flip the fraction). Doing this first lets us finish with a positive index.27−1/3 = (127)1/3
- Take the cube root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 3 times gives the base?cube root of 127: the cube root of the top and of the bottom: 1 and 3, so it is 13.
- The numerator is 1, so there is no power to apply. The answer is just the root.13
Check. Check with the other order: raise to the power first, then take the root. 1271 = 127 and its cube root is 13. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.16−3/4: denominator 4 means the fourth root; numerator 3 means the power 3; the minus sign means the reciprocal
- Deal with the minus sign: take the reciprocal first. A negative index means "one over" (or flip the fraction). Doing this first lets us finish with a positive index.16−3/4 = (116)3/4
- Take the fourth root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 4 times gives the base?fourth root of 116: the fourth root of the top and of the bottom: 1 and 2, so it is 12.
- Raise the answer to the power 3. The numerator is the power.(12)3 = 18 = 18
Check. Check with the other order: raise to the power first, then take the root. 1163 = 14096 and its fourth root is 18. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.25−3/2: denominator 2 means the square root; numerator 3 means the power 3; the minus sign means the reciprocal
- Deal with the minus sign: take the reciprocal first. A negative index means "one over" (or flip the fraction). Doing this first lets us finish with a positive index.25−3/2 = (125)3/2
- Take the square root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself once more gives the base?square root of 125: the square root of the top and of the bottom: 1 and 5, so it is 15.
- Raise the answer to the power 3. The numerator is the power.(15)3 = 1125 = 1125
Check. Check with the other order: raise to the power first, then take the root. 1253 = 115,625 and its square root is 1125. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.32−2/5: denominator 5 means the fifth root; numerator 2 means the power 2; the minus sign means the reciprocal
- Deal with the minus sign: take the reciprocal first. A negative index means "one over" (or flip the fraction). Doing this first lets us finish with a positive index.32−2/5 = (132)2/5
- Take the fifth root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 5 times gives the base?fifth root of 132: the fifth root of the top and of the bottom: 1 and 2, so it is 12.
- Raise the answer to the power 2. The numerator is the power.(12)2 = 14 = 14
Check. Check with the other order: raise to the power first, then take the root. 1322 = 11024 and its fifth root is 14. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.(49)1/2: denominator 2 means the square root; numerator 1 means the power 1
- Take the square root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself once more gives the base?square root of 49: the square root of the top and of the bottom: 2 and 3, so it is 23.
- The numerator is 1, so there is no power to apply. The answer is just the root.23
Check. Check with the other order: raise to the power first, then take the root. 491 = 49 and its square root is 23. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.(827)2/3: denominator 3 means the cube root; numerator 2 means the power 2
- Take the cube root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 3 times gives the base?cube root of 827: the cube root of the top and of the bottom: 2 and 3, so it is 23.
- Raise the answer to the power 2. The numerator is the power.(23)2 = 49 = 49
Check. Check with the other order: raise to the power first, then take the root. 8272 = 64729 and its cube root is 49. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.(278)−2/3: denominator 3 means the cube root; numerator 2 means the power 2; the minus sign means the reciprocal
- Deal with the minus sign: take the reciprocal first. A negative index means "one over" (or flip the fraction). Doing this first lets us finish with a positive index.(278)−2/3 = (827)2/3
- Take the cube root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 3 times gives the base?cube root of 827: the cube root of the top and of the bottom: 2 and 3, so it is 23.
- Raise the answer to the power 2. The numerator is the power.(23)2 = 49 = 49
Check. Check with the other order: raise to the power first, then take the root. 8272 = 64729 and its cube root is 49. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.(1681)−3/4: denominator 4 means the fourth root; numerator 3 means the power 3; the minus sign means the reciprocal
- Deal with the minus sign: take the reciprocal first. A negative index means "one over" (or flip the fraction). Doing this first lets us finish with a positive index.(1681)−3/4 = (8116)3/4
- Take the fourth root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself 4 times gives the base?fourth root of 8116: the fourth root of the top and of the bottom: 3 and 2, so it is 32.
- Raise the answer to the power 3. The numerator is the power.(32)3 = 278 = 278
Check. Check with the other order: raise to the power first, then take the root. 81163 = 531,4414096 and its fourth root is 278. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Evaluate each fractional power, then do the multiplication or division.
- Work out each part separately. Use root first, then power, for each fractional index.81/3 = 2 (because 2 × 2 × 2 = 8)
41/2 = 2 (because 2 × 2 = 4) - Combine the values. Multiply the two values: 2 × 2.4
Check. Each part was checked by raising the root to the power: the working above shows the calculation. ✓
Watch out. Root first, then power.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Evaluate each fractional power, then do the multiplication or division.
- Work out each part separately. Use root first, then power, for each fractional index.272/3 = (3√27)2 = 32 = 9
91/2 = 3 - Combine the values. Divide: 9 ÷ 3.3
Check. Each part was checked by raising the root to the power: the working above shows the calculation. ✓
Watch out. Root first, then power.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Evaluate each fractional power, then do the multiplication or division.
- Work out each part separately. Use root first, then power, for each fractional index.161/4 = 2 (because 24 = 16)
2−1 = 1/2 - Combine the values. Add: 2 + 1/2.52
Check. Each part was checked by raising the root to the power: the working above shows the calculation. ✓
Watch out. Root first, then power.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Use a1/2 = √a.
- Read the index. The index is 1/2, so the denominator 2 means the square root.x1/2
- Write as a root. √x
Check. √x × √x = x and x1/2 × x1/2 = x1. ✓
Watch out. The denominator is the root.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Use n√a = a1/n.
- Identify the root. A cube root has n = 3.3√y
- Write as a power. Index 1/3.y1/3
Check. (y1/3)3 = y1. ✓
Watch out. Cube root means index 1/3.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Denominator is the root, numerator is the power.
- Read the index 2/3. The denominator 3 is the cube root; the numerator 2 is the power.52/3
- Write it. (3√5)2, which also equals 3√(52) = 3√25
Check. Both forms mean “cube root and square”, in either order. ✓
Watch out. Do not forget the numerator.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Use the meaning of the index 1/2.
- See what Eve did. 9 ÷ 2 = 4.5, so she halved 9.Eve: 91/2 = 9 ÷ 2 = 4.5
- Use the correct meaning. An index of 1/2 means square root.91/2 = √9 = 3
- Check. 3 × 3 = 9.3
Check. 91/2 × 91/2 = 91 = 9 and 3 × 3 = 9. ✓
Watch out. A fractional index is a root, not a division.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Evaluate each power exactly, then multiply.
- Work out 82/3. Cube root first, then square.3√8 = 2, and 22 = 4
- Work out 4−1/2. Negative index: reciprocal. Then the square root.4−1/2 = 141/2 = 12
- Multiply. 4 × 12 = 2
Check. Write both as powers of 2: 82/3 = 22, 4−1/2 = 2−1, product 21 = 2. ✓
Watch out. Show each value before multiplying.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.43/2: denominator 2 means the square root; numerator 3 means the power 3
- Take the square root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself once more gives the base?square root of 4: 22 = 4, so it is 2.
- Raise the answer to the power 3. The numerator is the power.23 = 2 × 2 × 2 = 8
Check. Check with the other order: raise to the power first, then take the root. 43 = 64 and its square root is 8. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.93/2: denominator 2 means the square root; numerator 3 means the power 3
- Take the square root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself once more gives the base?square root of 9: 32 = 9, so it is 3.
- Raise the answer to the power 3. The numerator is the power.33 = 3 × 3 × 3 = 27
Check. Check with the other order: raise to the power first, then take the root. 93 = 729 and its square root is 27. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A root undoes a power: the square root of 49 is 7 because 72 = 49.
Method. Fractional index: the denominator is the root, the numerator is the power. am/n = (n√a)m. Do the root first.
- Read the fractional index. The denominator of the index tells you which root to take. The numerator tells you which power to raise to. For example 81/3 is the cube root of 8, because (81/3)3 = 81 = 8.253/2: denominator 2 means the square root; numerator 3 means the power 3
- Take the square root first. Doing the root first keeps the numbers small. Ask: which number multiplied by itself once more gives the base?square root of 25: 52 = 25, so it is 5.
- Raise the answer to the power 3. The numerator is the power.53 = 5 × 5 × 5 = 125
Check. Check with the other order: raise to the power first, then take the root. 253 = 15,625 and its square root is 125. ✓
Watch out. a1/2 is the square root of a. It is not a divided by 2 or a × one half.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Multiply 491/2 by itself with the multiplication law.
- Multiply the power by itself. Same base, so add the indices.491/2 × 491/2 = 491/2+1/2 = 491
- Simplify. An index of 1 changes nothing.491 = 49
- Conclude. A positive number that multiplies by itself to give 49 is the square root of 49.491/2 = √49 = 7
Check. 7 × 7 = 49. ✓
Watch out. This is why the index 1/2 means square root.
Exercise 6H Fractional indices in algebra and equations
Non-calculatorKey words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x3 the coefficient is 7. A letter on its own, like x, really means x1.
Method. Multiply the numbers, then add the indices of each letter: am × an = am+n.
- Deal with the letter x: add the indices. The letter is the same in both terms, so the index law applies. (x on its own means x1.)x1/2 × x3/2 = x1/2 + 3/2 = x2
- Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.x2
Check. Choose numbers: x = 4. The original expression has the value 16, and the answer x2 also has the value 16. ✓
Watch out. Add the indices of the same letter only. You cannot combine different letters: a2 × b3 stays as it is.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x3 the coefficient is 7. A letter on its own, like x, really means x1.
Method. Multiply the numbers, then add the indices of each letter: am × an = am+n.
- Deal with the letter x: add the indices. The letter is the same in both terms, so the index law applies. (x on its own means x1.)x2/3 × x1/3 = x2/3 + 1/3 = x1
- Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.x
Check. Choose numbers: x = 4. The original expression has the value 4, and the answer x also has the value 4. ✓
Watch out. Add the indices of the same letter only. You cannot combine different letters: a2 × b3 stays as it is.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x3 the coefficient is 7. A letter on its own, like x, really means x1.
Method. Divide the numbers, then subtract the indices of each letter: am ÷ an = am−n.
- Deal with the letter y: subtract the indices. The letter is the same in both terms, so the index law applies. (y on its own means y1.)y5/2 ÷ y1/2 = y5/2 − 1/2 = y2
- Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.y2
Check. Choose numbers: y = 4. The original expression has the value 16, and the answer y2 also has the value 16. ✓
Watch out. Subtract the indices (top minus bottom), and divide the numbers. Do not subtract the numbers.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x3 the coefficient is 7. A letter on its own, like x, really means x1.
Method. Raise the number and every letter to the power. Multiply the indices: (am)n = am×n.
- Raise the letter a to the power: multiply the indices. (am)n = am×n.(a1/2)4 = a1/2 × 4 = a2
- Put it all together. a2
Check. Choose numbers: a = 4. The original expression has the value 16, and the answer a2 also has the value 16. ✓
Watch out. The number in front is raised to the power too: (2a)3 = 8a3, not 2a3.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x3 the coefficient is 7. A letter on its own, like x, really means x1.
Method. Raise the number and every letter to the power. Multiply the indices: (am)n = am×n.
- Raise the number in front to the power. Everything inside the bracket is raised to the power, and that includes the number.The 4th root of 16 is 2, because 24 = 16.
Then 23 = 8 - Raise the letter x to the power: multiply the indices. (am)n = am×n.(x8)3/4 = x8 × 3/4 = x6
- Put it all together. 8x6
Check. Choose numbers: x = 4. The original expression has the value 32,768, and the answer 8x6 also has the value 32,768. ✓
Watch out. The number in front is raised to the power too: (2a)3 = 8a3, not 2a3.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x3 the coefficient is 7. A letter on its own, like x, really means x1.
Method. Raise the number and every letter to the power. Multiply the indices: (am)n = am×n.
- Raise the number in front to the power. Everything inside the bracket is raised to the power, and that includes the number.The cube root of 27 is 3, because 33 = 27.
Then 31 = 3 - Raise the letter y to the power: multiply the indices. (am)n = am×n.(y6)1/3 = y6 × 1/3 = y2
- Put it all together. 3y2
Check. Choose numbers: y = 4. The original expression has the value 48, and the answer 3y2 also has the value 48. ✓
Watch out. The number in front is raised to the power too: (2a)3 = 8a3, not 2a3.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x3 the coefficient is 7. A letter on its own, like x, really means x1.
Method. Raise the number and every letter to the power. Multiply the indices: (am)n = am×n.
- Raise the number in front to the power. Everything inside the bracket is raised to the power, and that includes the number.The square root of 4 is 2, because 22 = 4.
Then 21 = 2 - Raise the letter a to the power: multiply the indices. (am)n = am×n.(a2)1/2 = a2 × 1/2 = a1
- Put it all together. 2a
Check. Choose numbers: a = 4. The original expression has the value 8, and the answer 2a also has the value 8. ✓
Watch out. The number in front is raised to the power too: (2a)3 = 8a3, not 2a3.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x3 the coefficient is 7. A letter on its own, like x, really means x1.
Method. Raise the number and every letter to the power. Multiply the indices: (am)n = am×n.
- Raise the number in front to the power. Everything inside the bracket is raised to the power, and that includes the number.The cube root of 8 is 2, because 23 = 8.
Then 22 = 4 - Raise the letter x to the power: multiply the indices. (am)n = am×n.(x3)2/3 = x3 × 2/3 = x2
- Put it all together. 4x2
Check. Choose numbers: x = 4. The original expression has the value 64, and the answer 4x2 also has the value 64. ✓
Watch out. The number in front is raised to the power too: (2a)3 = 8a3, not 2a3.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x3 the coefficient is 7. A letter on its own, like x, really means x1.
Method. Raise the number and every letter to the power. Multiply the indices: (am)n = am×n.
- Raise the number in front to the power. Everything inside the bracket is raised to the power, and that includes the number.The 4th root of 81 is 3, because 34 = 81.
Then 31 = 3 - Raise the letter a to the power: multiply the indices. (am)n = am×n.(a4)1/4 = a4 × 1/4 = a1
- Raise the letter b to the power: multiply the indices. (am)n = am×n.(b8)1/4 = b8 × 1/4 = b2
- Put it all together. 3ab2
Check. Choose numbers: a = 4, b = 9. The original expression has the value 972, and the answer 3ab2 also has the value 972. ✓
Watch out. The number in front is raised to the power too: (2a)3 = 8a3, not 2a3.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x3 the coefficient is 7. A letter on its own, like x, really means x1.
Method. Raise the number and every letter to the power. Multiply the indices: (am)n = am×n.
- Raise the number in front to the power. Everything inside the bracket is raised to the power, and that includes the number.The square root of 9 is 3, because 32 = 9.
Then 3-1 = 13 - Raise the letter x to the power: multiply the indices. (am)n = am×n.(x4)−1/2 = x4 × (−1/2) = x−2
- Put it all together. 13x−2
- Write with positive indices. A negative index means "one over": a−n = 1/an.13x−2 = 13x2
Check. Choose numbers: x = 4. The original expression has the value 148, and the answer 13x−2 also has the value 148. ✓
Watch out. The number in front is raised to the power too: (2a)3 = 8a3, not 2a3.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x3 the coefficient is 7. A letter on its own, like x, really means x1.
Method. Raise the number and every letter to the power. Multiply the indices: (am)n = am×n.
- Raise the number in front to the power. Everything inside the bracket is raised to the power, and that includes the number.The cube root of 8 is 2, because 23 = 8.
Then 2-1 = 12 - Raise the letter y to the power: multiply the indices. (am)n = am×n.(y6)−1/3 = y6 × (−1/3) = y−2
- Put it all together. 12y−2
- Write with positive indices. A negative index means "one over": a−n = 1/an.12y−2 = 12y2
Check. Choose numbers: y = 4. The original expression has the value 132, and the answer 12y−2 also has the value 132. ✓
Watch out. The number in front is raised to the power too: (2a)3 = 8a3, not 2a3.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x3 the coefficient is 7. A letter on its own, like x, really means x1.
Method. Divide the numbers, then subtract the indices of each letter: am ÷ an = am−n.
- Deal with the letter x: subtract the indices. The letter is the same in both terms, so the index law applies. (x on its own means x1.)x1/2 ÷ x3/2 = x1/2 − 3/2 = x−1 = 1x
- Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.x−1
- Write with positive indices. A negative index means "one over": a−n = 1/an.x−1 = 1x
Check. Choose numbers: x = 4. The original expression has the value 14, and the answer x−1 also has the value 14. ✓
Watch out. Subtract the indices (top minus bottom), and divide the numbers. Do not subtract the numbers.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x3 the coefficient is 7. A letter on its own, like x, really means x1.
Method. Multiply the numbers, then add the indices of each letter: am × an = am+n.
- Multiply the numbers in front (the coefficients). The numbers can be dealt with separately from the letters.2 × 3 = 6
- Deal with the letter x: add the indices. The letter is the same in both terms, so the index law applies. (x on its own means x1.)x1/2 × x3/2 = x1/2 + 3/2 = x2
- Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.6x2
Check. Choose numbers: x = 4. The original expression has the value 96, and the answer 6x2 also has the value 96. ✓
Watch out. Add the indices of the same letter only. You cannot combine different letters: a2 × b3 stays as it is.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x3 the coefficient is 7. A letter on its own, like x, really means x1.
Method. Work through the expression one operation at a time, using the laws of indices on the number and on each letter.
- Look at the order of the operations. Work from the left, one operation at a time, exactly as in BIDMAS (brackets and powers first).
- Do the first operation: raise to the power 1/2. Raise the number and every index to the power: multiply the indices.The square root of 25 is 5, because 52 = 25.
Then 51 = 5
(a4)1/2 = a4 × 1/2 = a2
Result so far: 5a2 - Do the second operation: multiply by 2a−1. Multiply the numbers; add the indices of each letter.5 × 2 = 10
a2 × a−1 = a2 + (−1) = a1
Result so far: 10a
Check. Choose numbers: a = 2. The original expression gives 20 and the answer 10a gives 20. ✓
Watch out. Keep the working in the same order each time. Deal with brackets and powers first, then multiplication and division from left to right.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81. A coefficient is the number in front of a letter, so in 7x3 the coefficient is 7. A letter on its own, like x, really means x1.
Method. Work through the expression one operation at a time, using the laws of indices on the number and on each letter.
- Look at the order of the operations. Work from the left, one operation at a time, exactly as in BIDMAS (brackets and powers first).
- Do the first operation: raise to the power 3/2. Raise the number and every index to the power: multiply the indices.The square root of 16 is 4, because 42 = 16.
Then 43 = 64
(x2)3/2 = x2 × 3/2 = x3
(y4)3/2 = y4 × 3/2 = y6
Result so far: 64x3y6 - Do the second operation: divide by 8xy3. Divide the numbers; subtract the indices of each letter.64 ÷ 8 = 8
x3 ÷ x1 = x3 − 1 = x2
y6 ÷ y3 = y6 − 3 = y3
Result so far: 8x2y3
Check. Choose numbers: x = 2, y = 3. The original expression gives 864 and the answer 8x2y3 gives 864. ✓
Watch out. Keep the working in the same order each time. Deal with brackets and powers first, then multiplication and division from left to right.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Undo a fractional power by raising both sides to the reciprocal power.
- Spot the fractional index. The letter is raised to the power p/q. To get the letter on its own we need to undo that power.x3/2 = 27
- Raise both sides to the reciprocal power. The reciprocal of 3/2 is 2/3. Raising to this power gives (x3/2)2/3 = x1, which is just x. Do the same to the right-hand side to keep the equation balanced.x = 272/3
- Evaluate: root first, then power. The denominator 3 is the root and the numerator 2 is the power.the cube root of 27 is 3, then 32 = 9
x = 9
Check. Put x = 9 back: the square root of 9 is 3, and 33 = 27. ✓
Watch out. Use the reciprocal of the index (flip the fraction), not the negative of it.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Undo a fractional power by raising both sides to the reciprocal power.
- Spot the fractional index. The letter is raised to the power p/q. To get the letter on its own we need to undo that power.x2/3 = 4
- Raise both sides to the reciprocal power. The reciprocal of 2/3 is 3/2. Raising to this power gives (x2/3)3/2 = x1, which is just x. Do the same to the right-hand side to keep the equation balanced.x = 43/2
- Evaluate: root first, then power. The denominator 2 is the root and the numerator 3 is the power.the square root of 4 is 2, then 23 = 8
x = 8
Check. Put x = 8 back: the cube root of 8 is 2, and 22 = 4. ✓
Watch out. Use the reciprocal of the index (flip the fraction), not the negative of it.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Undo a fractional power by raising both sides to the reciprocal power.
- Spot the fractional index. The letter is raised to the power p/q. To get the letter on its own we need to undo that power.x3/4 = 8
- Raise both sides to the reciprocal power. The reciprocal of 3/4 is 4/3. Raising to this power gives (x3/4)4/3 = x1, which is just x. Do the same to the right-hand side to keep the equation balanced.x = 84/3
- Evaluate: root first, then power. The denominator 3 is the root and the numerator 4 is the power.the cube root of 8 is 2, then 24 = 16
x = 16
Check. Put x = 16 back: the fourth root of 16 is 2, and 23 = 8. ✓
Watch out. Use the reciprocal of the index (flip the fraction), not the negative of it.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Undo a fractional power by raising both sides to the reciprocal power.
- Spot the fractional index. The letter is raised to the power p/q. To get the letter on its own we need to undo that power.x1/2 = 7
- Raise both sides to the reciprocal power. The reciprocal of 1/2 is 2/1. Raising to this power gives (x1/2)2/1 = x1, which is just x. Do the same to the right-hand side to keep the equation balanced.x = 72/1
- Evaluate: root first, then power. The denominator 1 is the root and the numerator 2 is the power.the 1th root of 7 is 7, then 72 = 49
x = 49
Check. Put x = 49 back: the square root of 49 is 7, and 71 = 7. ✓
Watch out. Use the reciprocal of the index (flip the fraction), not the negative of it.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Undo a fractional power by raising both sides to the reciprocal power.
- Spot the fractional index. The letter is raised to the power p/q. To get the letter on its own we need to undo that power.x1/3 = 4
- Raise both sides to the reciprocal power. The reciprocal of 1/3 is 3/1. Raising to this power gives (x1/3)3/1 = x1, which is just x. Do the same to the right-hand side to keep the equation balanced.x = 43/1
- Evaluate: root first, then power. The denominator 1 is the root and the numerator 3 is the power.the 1th root of 4 is 4, then 43 = 64
x = 64
Check. Put x = 64 back: the cube root of 64 is 4, and 41 = 4. ✓
Watch out. Use the reciprocal of the index (flip the fraction), not the negative of it.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Write both sides as powers of the same base, then compare the indices.
- Write both sides as powers of the same base. The bases 4 and 8 look different, but both are powers of 2.4 = 22
8 = 23 - Replace them in the equation. Use the power-of-a-power law: (am)n = am×n. So raising 22 to the power x multiplies the indices.(22)x = 22x, so 22x = 23
- Equate the indices. The bases are the same, so the indices must be equal.2x = 3
- Solve. Divide both sides by 2.x = 32 = 32
Check. Put x = 32 back: 43/2 = (22)3/2 = 22 × (3/2) = 23 = 8. ✓
Watch out. Find the common base first (2, 3 or 5 are the usual ones). Then the powers-of-a-power law turns the problem into a simple equation.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Write both sides as powers of the same base, then compare the indices.
- Write both sides as powers of the same base. The bases 9 and 27 look different, but both are powers of 3.9 = 32
27 = 33 - Replace them in the equation. Use the power-of-a-power law: (am)n = am×n. So raising 32 to the power x multiplies the indices.(32)x = 32x, so 32x = 33
- Equate the indices. The bases are the same, so the indices must be equal.2x = 3
- Solve. Divide both sides by 2.x = 32 = 32
Check. Put x = 32 back: 93/2 = (32)3/2 = 32 × (3/2) = 33 = 27. ✓
Watch out. Find the common base first (2, 3 or 5 are the usual ones). Then the powers-of-a-power law turns the problem into a simple equation.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Write both sides as powers of the same base, then compare the indices.
- Write both sides as powers of the same base. The bases 8 and 4 look different, but both are powers of 2.8 = 23
4 = 22 - Replace them in the equation. Use the power-of-a-power law: (am)n = am×n. So raising 23 to the power x multiplies the indices.(23)x = 23x, so 23x = 22
- Equate the indices. The bases are the same, so the indices must be equal.3x = 2
- Solve. Divide both sides by 3.x = 23 = 23
Check. Put x = 23 back: 82/3 = (23)2/3 = 23 × (2/3) = 22 = 4. ✓
Watch out. Find the common base first (2, 3 or 5 are the usual ones). Then the powers-of-a-power law turns the problem into a simple equation.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Write both sides as powers of the same base, then compare the indices.
- Write both sides as powers of the same base. The bases 25 and 5 look different, but both are powers of 5.25 = 52
5 = 51 - Replace them in the equation. Use the power-of-a-power law: (am)n = am×n. So raising 52 to the power x multiplies the indices.(52)x = 52x, so 52x = 51
- Equate the indices. The bases are the same, so the indices must be equal.2x = 1
- Solve. Divide both sides by 2.x = 12 = 12
Check. Put x = 12 back: 251/2 = (52)1/2 = 52 × (1/2) = 51 = 5. ✓
Watch out. Find the common base first (2, 3 or 5 are the usual ones). Then the powers-of-a-power law turns the problem into a simple equation.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Write both sides as powers of the same base, then compare the indices.
- Write both sides as powers of the same base. The bases 9 and 127 look different, but both are powers of 3.9 = 32
127 = 127 = 3−3 - Replace them in the equation. Use the power-of-a-power law: (am)n = am×n. So raising 32 to the power x multiplies the indices.(32)x = 32x, so 32x = 3−3
- Equate the indices. The bases are the same, so the indices must be equal.2x = −3
- Solve. Divide both sides by 2.x = −32 = −32
Check. Put x = −32 back: 9−3/2 = (32)−3/2 = 32 × (−3/2) = 3−3 = 127. ✓
Watch out. Find the common base first (2, 3 or 5 are the usual ones). Then the powers-of-a-power law turns the problem into a simple equation.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Write both sides as powers of the same base, then compare the indices.
- Write both sides as powers of the same base. The bases 4 and 18 look different, but both are powers of 2.4 = 22
18 = 18 = 2−3 - Replace them in the equation. Use the power-of-a-power law: (am)n = am×n. So raising 22 to the power x multiplies the indices.(22)x = 22x, so 22x = 2−3
- Equate the indices. The bases are the same, so the indices must be equal.2x = −3
- Solve. Divide both sides by 2.x = −32 = −32
Check. Put x = −32 back: 4−3/2 = (22)−3/2 = 22 × (−3/2) = 2−3 = 18. ✓
Watch out. Find the common base first (2, 3 or 5 are the usual ones). Then the powers-of-a-power law turns the problem into a simple equation.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Expand the bracket as (a + b)2 = a2 + 2ab + b2, with a = x1/2 and b = x−1/2.
- Square the first term. (x1/2)2: multiply the indices.(x1/2)2 = x1 = x
- Find twice the product. Add the indices when multiplying.2 × x1/2 × x−1/2 = 2x0 = 2
- Square the second term. (x−1/2)2 = x−1 = 1x
- Add the three parts. x + 2 + 1x
Check. Try x = 4: (2 + 1/2)2 = 6.25 and 4 + 2 + 1/4 = 6.25. ✓
Watch out. Write out (a + b)(a + b) if you are unsure.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Express x in terms of p first, then square it.
- Undo the cube root. Raise both sides of p = x1/3 to the power 3.p3 = (x1/3)3 = x
- Square x. Power of a power: multiply the indices.x2 = (p3)2 = p6
Check. Try x = 8: p = 2, x2 = 64 and p6 = 64. ✓
Watch out. Cubing undoes the index 1/3.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Write both sides as powers of 2, then compare the indices.
- Write 8 as a power of 2. 8 = 2 × 2 × 2.8 = 23
- Use the power-of-a-power law on the right. Multiply the indices: (23)x−1 = 23(x−1).22x = 23x − 3
- Equate the indices. The bases are the same.2x = 3x − 3
- Solve. Subtract 2x from both sides.0 = x − 3, so x = 3
Check. Left: 26 = 64. Right: 82 = 64. ✓
Watch out. Bracket the whole of the index x − 1 when multiplying by 3.
Key words. In 34, the base is 3 and the index (also called the power or exponent) is 4. The index tells you how many lots of the base are multiplied together: 34 = 3 × 3 × 3 × 3 = 81.
Method. Write everything as a power of 2.
- Write 4 and 1024 as powers of 2. 4 = 22 and 1024 = 210.4x = 22x
- Multiply: add the indices. 22x × 2x+1 = 23x+1
- Equate the indices. 23x+1 = 210.3x + 1 = 10
- Solve. 3x = 9, so x = 3
Check. 43 × 24 = 64 × 16 = 1024. ✓
Watch out. Choose the smallest common base.