GCSE Maths: Roots and fractional indices (Foundation)
Part of the GCSE Maths standard form, indices and surds chapter (Foundation). Chapter overview · Higher 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.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. 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.