GCSE Maths Standard Form, Indices and Surds (Higher): check-up and exam-style questions

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 45,000 is 4.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 4.5
3. Count how many places the decimal point has moved. Starting from 45000 the decimal point has to move 4 places to the left to sit after the 4. Each place moved is a factor of 10.
   45000 → 4.5: 4 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 4
5. Write the answer. Put A and the power of 10 together.
   4.5 × 10⁴

Check. Convert back: 4.5 × 10⁴ = 45000 = 45,000. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is −3, which is negative, so the number is small. Dividing by 10 moves every digit one place to the right, which is the same as moving the decimal point one place to the left.
   10⁻³ means divide by 10, 3 times
2. Move the decimal point 3 places to the left. Move it one place at a time and fill the new places with zeros.
   3.2 → 0.32 → 0.032 → 0.0032
3. Write the answer. The last number in the chain is the answer.
   0.0032

Check. Convert back to standard form: 0.0032 has first non-zero digit 3, so A = 3.2 and n = −3. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

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 laws of indices in order: add indices to multiply, subtract indices to divide.

1. Check that every base is the same. The laws of indices only work when the bases match.
   Every base is 2.
2. Work out the top first: multiply, so add the indices. a^m × aⁿ = a^m+n.
   5 + 3 = 8, so the top is 2⁸
3. Divide: subtract the indices. a^m ÷ aⁿ = a^m−n.
   2⁸2⁴ = 2⁸ ÷ 2⁴ = 2⁴, because 8 − 4 = 4

Check. The total of the indices on top is 8 and on the bottom is 4; 8 − 4 = 4. ✓

Watch out. Deal with the multiplication in the top (and bottom) first, then divide. Only subtract once the top and bottom are each a single 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. A negative index means "one over": a⁻ⁿ = 1 ÷ aⁿ. For a fraction, flip it upside down and make the index positive.

1. Spot the negative index. A negative index does not make the answer negative. It means "one over".
   5⁻²: the index is −2
2. Turn it into a reciprocal with a positive index. The rule is a⁻ⁿ = 1aⁿ. (Check with a pattern: 5² = 25, 5¹ = 5, 5⁰ = 1, and each step is ÷ 5, so 5⁻¹ = 15.)
   5⁻² = 15²
3. Work out the bottom. 5² means 2 lots of 5 multiplied together.
   5² = 5 × 5 = 25
4. Write the answer as a fraction.
   5⁻² = 125

Check. Multiply the answer by 5²: 125 × 25 = 1, and 5⁻² × 5² = 5⁰ = 1. ✓

Watch out. A negative index does not give a negative answer: 5⁻² = 125, not a negative number.

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: a^m × aⁿ = a^m+n.

1. Multiply the numbers in front (the coefficients). The numbers can be dealt with separately from the letters.
   3 × 4 = 12
2. 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 × x³ = x^{2 + 3} = x⁵
3. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   12x⁵

Check. Choose numbers: x = 2. The original expression has the value 384, and the answer 12x⁵ also has the value 384. ✓

Watch out. Add the indices of the same letter only. You cannot combine different letters: a² × b³ stays as it is.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Multiply the front numbers, add the powers of 10, then make sure the front number is between 1 and 10.

1. Write the multiplication with the number parts together and the powers of 10 together. Multiplication can be done in any order, so we can group the front numbers and group the powers of 10.
   (2 × 10³) × (4 × 10⁵) = (2 × 4) × (10³ × 10⁵)
2. Multiply the front numbers. Do this like an ordinary decimal multiplication.
   2 × 4 = 8
3. Multiply the powers of 10: add the indices. 10^a × 10^b = 10^a+b.
   10³ × 10⁵ = 10^{3 + 5} = 10⁸
4. Put the two parts together.
   8 × 10⁸
5. Write the answer.
   8 × 10⁸

Check. As ordinary numbers: 2000 × 400000 = 800000000, which is 8 × 10⁸. ✓

Watch out. Add the powers when you multiply (do not multiply them). Always check the front number at the end: 12 × 10⁷ is not in standard form.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Put the numbers into the formula but leave π as the symbol π. Then the answer is exact.

1. Write the formula for the area of a circle. The area of a circle is π times the radius squared.
   A = πr²
2. Substitute the radius. Replace r with the radius. Leave π as the symbol.
   A = π × 5²
3. Work out the square first. In BIDMAS, powers come before multiplication. Square the radius, not the π.
   5² = 5 × 5 = 25
4. Write the number in front of π. Write the exact answer with the number first and then π, and add the units.
   25π cm²

Check. Use the π button to get a decimal and compare: 25π = 78.54 to 3 decimal places, so about 78.5 cm² to 1 d.p. This is similar in size to 78.5 found using 3.14. ✓

Watch out. Do not replace π by 3.14 or press the π key if the question says "in terms of π" or "give your answer exactly". Also square only the radius.

Method. Work out each root or power separately, then add and subtract.

1. 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
   ³√8 = 2, because 2 × 2 × 2 = 8
2. Put the values into the calculation. Keep the signs the same.
   √49 + ³√8 = 7 + 2
3. Work out the answer. Add and subtract from left to right.
   7 + 2 = 9

Check. Work through it again from the left: 7 + 2 = 9. ✓

Watch out. Do each root or power first (BIDMAS: powers and roots before adding and subtracting).

Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.

Method. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.

1. List the square numbers and see which ones divide into 48. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.
   4: 48 ÷ 4 = 12 ✓ goes in exactly
   9: 48 ÷ 9 = 5 remainder 3 ✗ does not go in exactly
   16: 48 ÷ 16 = 3 ✓ goes in exactly
   25: 48 ÷ 25 = 1 remainder 23 ✗ does not go in exactly
   36: 48 ÷ 36 = 1 remainder 12 ✗ does not go in exactly
2. Write 48 as 16 × 3. 16 is the largest square number that divides into 48. Use it so that the answer is fully simplified.
   48 = 16 × 3, so √48 = √16 × 3
3. Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.
   √16 × 3 = √16 × √3
4. Work out √16. 16 is a square number, so its root is a whole number: 4 × 4 = 16.
   √16 = 4, so √16 × √3 = 4√3
5. Write the answer.
   4√3

Check. Square the answer: (4√3)² = 4² × 3 = 16 × 3 = 48, and the original squared is (√48)² = 48. ✓

Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.

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. a^m/n = (ⁿ√a)^m. Do the root first.

1. 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 8^1/3 is the cube root of 8, because (8^1/3)³ = 8¹ = 8.
   8^2/3: denominator 3 means the cube root; numerator 2 means the power 2
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.
3. 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. a^1/2 is the square root of a. It is not a divided by 2 or a × one half.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 7,300,000 is 7.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 7.3
3. Count how many places the decimal point has moved. Starting from 7300000 the decimal point has to move 6 places to the left to sit after the 7. Each place moved is a factor of 10.
   7300000 → 7.3: 6 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 6
5. Write the answer. Put A and the power of 10 together.
   7.3 × 10⁶

Check. Convert back: 7.3 × 10⁶ = 7300000 = 7,300,000. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is −4, which is negative, so the number is small. Dividing by 10 moves every digit one place to the right, which is the same as moving the decimal point one place to the left.
   10⁻⁴ means divide by 10, 4 times
2. Move the decimal point 4 places to the left. Move it one place at a time and fill the new places with zeros.
   4.5 → 0.45 → 0.045 → 0.0045 → 0.00045
3. Write the answer. The last number in the chain is the answer.
   0.00045

Check. Convert back to standard form: 0.00045 has first non-zero digit 4, so A = 4.5 and n = −4. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

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. When you multiply powers with the same base, you add the indices: a^m × aⁿ = a^m+n.

1. Check that the bases are the same. These laws only work when the base (the big number) is the same in both powers.
   Both bases are 2.
2. Write each power out in full. This shows what is really happening: we are counting how many lots of the base are multiplied.
   2³ = 2 × 2 × 2
   2⁴ = 2 × 2 × 2 × 2
   Together: 3 lots of 2 and 4 lots of 2 make 7 lots of 2.
3. Add the indices. Joining the lots means adding the number of lots.
   3 + 4 = 7
4. Write the answer as a single power. Keep the same base and use the new index.
   2³ × 2⁴ = 2⁷

Check. Work out the numbers: 2³ = 8 and 2⁴ = 16. 8 × 16 = 128, and 2⁷ = 128. ✓

Watch out. Add the indices when you multiply. Do not multiply the indices, and do not multiply the bases (3² × 3⁴ is not 9⁶).

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Multiply the front numbers, add the powers of 10, then make sure the front number is between 1 and 10.

1. Write the multiplication with the number parts together and the powers of 10 together. Multiplication can be done in any order, so we can group the front numbers and group the powers of 10.
   (3.2 × 10⁴) × (2.5 × 10³) = (3.2 × 2.5) × (10⁴ × 10³)
2. Multiply the front numbers. Do this like an ordinary decimal multiplication.
   Ignore the decimal points: 32 × 25 = 800
   There are 2 decimal places altogether in 3.2 and 2.5, so the answer has 2 decimal places: 3.2 × 2.5 = 8
3. Multiply the powers of 10: add the indices. 10^a × 10^b = 10^a+b.
   10⁴ × 10³ = 10^{4 + 3} = 10⁷
4. Put the two parts together.
   8 × 10⁷
5. Write the answer.
   8 × 10⁷

Check. As ordinary numbers: 32000 × 2500 = 80000000, which is 8 × 10⁷. ✓

Watch out. Add the powers when you multiply (do not multiply them). Always check the front number at the end: 12 × 10⁷ is not in standard form.

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: a^m ÷ aⁿ = a^m−n.

1. Divide the numbers in front (the coefficients). The numbers can be dealt with separately from the letters.
   6 ÷ 2 = 3
2. Deal with the letter a: subtract the indices. The letter is the same in both terms, so the index law applies. (a on its own means a¹.)
   a⁵ ÷ a² = a^{5 − 2} = a³
3. Put the number and the letters together. Write the coefficient first, then the letters in alphabetical order.
   3a³

Check. Choose numbers: a = 2. The original expression has the value 24, and the answer 3a³ also has the value 24. ✓

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.

Method. A negative index means "one over": a⁻ⁿ = 1 ÷ aⁿ. For a fraction, flip it upside down and make the index positive.

1. Spot the negative index. A negative index does not make the answer negative. It means "one over".
   4⁻²: the index is −2
2. Turn it into a reciprocal with a positive index. The rule is a⁻ⁿ = 1aⁿ. (Check with a pattern: 4² = 16, 4¹ = 4, 4⁰ = 1, and each step is ÷ 4, so 4⁻¹ = 14.)
   4⁻² = 14²
3. Work out the bottom. 4² means 2 lots of 4 multiplied together.
   4² = 4 × 4 = 16
4. Write the answer as a fraction.
   4⁻² = 116

Check. Multiply the answer by 4²: 116 × 16 = 1, and 4⁻² × 4² = 4⁰ = 1. ✓

Watch out. A negative index does not give a negative answer: 4⁻² = 116, not a negative number.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Order by the power of 10 first; use the front number to break ties.

1. List the power of 10 of each number. The power of 10 is the main thing that decides the size.
   6.1 × 10³: power 3
   8.2 × 10⁻²: power −2
   5.9 × 10³: power 3
   1.7 × 10⁴: power 4
2. Order by the powers of 10. We need smallest to largest. Remember that −7 is smaller than −3.
   power −2, power 3, power 3, power 4
3. If two numbers have the same power, compare their front numbers. With equal powers, the bigger front number is the bigger number.
   power 3: 6.1 and 5.9 → smaller first is 5.9
4. Write the numbers in order.
   8.2 × 10⁻² ,  5.9 × 10³ ,  6.1 × 10³ ,  1.7 × 10⁴

Check. As ordinary numbers: 0.082, 5900, 6100, 17000 — these increase along the list. ✓

Watch out. A negative power is smaller than a positive power, and a larger negative power (like −8) is smaller than −2.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Split into a rectangle and a semicircle, find each area exactly, then add.

1. Split the shape into a rectangle and a semicircle. It is easier to find the areas of simple shapes and add them.
   area = rectangle + semicircle
2. Find the area of the rectangle. area = length × width
   8 × 3 = 24 cm²
3. Find the radius of the semicircle. The diameter of the semicircle is the width of the rectangle. The radius is half the diameter.
   radius = 8 ÷ 2 = 4 cm
4. Find the area of the semicircle. A semicircle is half a circle: half of πr².
   12 × π × 4² = 12 × π × 16 = 8π cm²
5. Add the two areas. Leave π in the answer. The whole numbers and the π part cannot be combined into one term.
   (24 + 8π) cm²

Check. As a decimal: 24 + 25.13 = 49.13 cm² (about). The rectangle alone is 24 cm², so this is a little bigger. ✓

Watch out. Use the radius (half the width) in the circle formula, not the whole width.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Use the ×10^x key on a calculator, then write the display in standard form to the accuracy asked for.

1. Decide what to do. Time = distance ÷ speed.
   time = (3.84 × 10⁸) ÷ (3 × 10⁸)
2. Enter the calculation using the ×10^x key. The ×10^x key (on some calculators it is EXP) types "× 10 to the power" in one go. For a negative power, press the (−) key before the power number. Brackets stop the calculator mixing the numbers up.
   Press: ( 3.84 [×10^x] 8 ) [÷] ( 3 [×10^x] 8 ) [=]
3. Read the answer from the display. Your calculator may show the answer as an ordinary number or in its own standard form style, such as 2.4E−3. The digits are what matter.
   Display: 1.28 × 10⁰ (about)
4. Round to 3 significant figures and write in standard form. Round the front number A to 3 significant figures: look at the digit after the last one you keep, and round up if it is 5 or more. The power of 10 stays the same unless the rounding makes A reach 10.
   1.28 × 10⁰ s

Check. Estimate by rounding each front number to the nearest whole number: (4 × 10⁸) ÷ (3 × 10⁸) ≈ 1 × 10⁰, which is close to our answer. ✓

Watch out. Use the ×10^x key, not separate × 10 ^ keys, unless you use brackets. Always check the power is sensible with a quick estimate.

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 with a cube root, then the area of one face, then multiply by 6.

1. Find the edge. Edge × edge × edge = 343, so edge = ³√343.
   ³√343 = 7, because 7 × 7 × 7 = 343
2. Find the area of one face. Each face is a square with side 7 cm.
   7 × 7 = 49 cm²
3. Multiply by the number of faces. A cube has 6 faces.
   6 × 49 = 294 cm²

Check. 7³ = 343 and 6 × 49 = 294. ✓

Watch out. Marks: 1 for the edge, 1 for the face, 1 for the surface area.

Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.

Method. Simplify every surd first. Then add or subtract the like surds (same number under the root).

1. Simplify each surd on its own. Surds can only be added or subtracted when the number under the root is the same (like terms). Simplifying each surd often makes them match.
   √75 = √25 × 3 = 5√3
   √12 = √4 × 3 = 2√3
2. Rewrite the expression with the simplified surds. Keep the signs the same as before.
   5√3 + 2√3
3. Collect the like surds. Like surds have the same number under the root. Add the numbers in front, just as you would add 3 apples and 5 apples. The root stays the same.
   √3 terms: 5 + 2 = 7, so 7√3
4. Write the answer.
   7√3

Check. Use decimals: the original is about 12.124 and the answer is about 12.124. They agree. ✓

Watch out. Only add or subtract surds that have the same number under the root. √2 + √3 is not √5.

Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.

Method. Expand the brackets by multiplying every term by every term. Simplify each product, then collect like terms.

1. Write the product. We multiply every term in the first bracket by every term in the second bracket. A way to remember the four products is First, Outer, Inner, Last (FOIL).
   (3 + √2)(3 − √2)
2. Multiply term by term. Simplify each product. Use √a × √a = a and √a × √b = √(ab). Take care with the signs.
   First: 3 × 3 = 9
   Outer: 3 × (−√2) = −3√2
   Inner: √2 × 3 = 3√2
   Last: √2 × (−√2) = −√4 = −2
3. Write all the simplified terms in one line. Keep the signs.
   9 − 3√2 + 3√2 − 2
4. Collect like terms. Add the whole numbers together. Add the like surds together (the same number under the root).
   7

Check. Decimals: the original is 4.414 × 1.586 ≈ 7.00 and the answer is about 7.00. ✓

Watch out. (2 + √3)² is not 4 + 3. Always write out both brackets: (2 + √3)(2 + √3). The middle terms do not disappear.

Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.

Method. Multiply the top and bottom by the surd in the denominator. This makes the denominator a whole number.

1. Say what is wrong with the fraction. A denominator (the bottom) containing a surd is not allowed in a fully simplified answer. Rationalising means rewriting the fraction so that the bottom is a whole number.
   12√6
2. Multiply the top and the bottom by √6. √6 ÷ √6 = 1, so multiplying by it does not change the value of the fraction. We choose √6 because √6 × √6 = 6, a whole number.
   12√6 × √6√6 = 12√6√6×√6
3. Work out the top and the bottom. Top: multiply each term by √6. Bottom: √6 × √6 = 6.
   12√66
4. Simplify the fraction. Divide the top and the bottom by any number that goes into every term.
   2√6

Check. Decimals: 4.899 before and 4.899 after. They agree. ✓

Watch out. You must multiply the top and the bottom. If you only multiply the bottom, you change the value of the fraction.

Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.

Method. Multiply the top and bottom by the conjugate of the bottom. The bottom becomes a whole number (difference of two squares).

1. Say what is wrong with the fraction. The bottom contains a surd. We need to rationalise it (make the bottom a whole number). Multiplying by the surd alone does not work here, because the bottom has two terms.
   13 + √5
2. Write down the conjugate of the bottom. The conjugate is the same two terms with the sign between them changed. Multiplying a bracket by its conjugate gives a difference of two squares, so the surd disappears.
   bottom: 3 + √5   conjugate: 3 − √5
3. Multiply top and bottom by the conjugate. Multiplying the top and the bottom by the same thing is multiplying by 1, so the value does not change.
   1(3 + √5) × (3 − √5)(3 − √5)
4. Expand the bottom using the difference of two squares. (p + q)(p − q) = p² − q², and the middle terms cancel.
   (3 + √5)(3 − √5) = 3² − (√5)² = 9 − 5 = 4
5. Expand the top. Multiply the top by the conjugate, term by term.
   1 × (3 − √5) = 3 − √5
6. Write the new fraction and simplify. Divide every term on the top by the number on the bottom if you can.
   3 − √54 = 3 − √54

Check. Decimals: 0.191 before and 0.191 after. They agree. ✓

Watch out. Change the sign between the two terms to get the conjugate, and multiply both the top and the bottom by 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. 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. a^m/n = (ⁿ√a)^m. Do the root first.

1. 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 8^1/3 is the cube root of 8, because (8^1/3)³ = 8¹ = 8.
   16^−3/4: denominator 4 means the fourth root; numerator 3 means the power 3; the minus sign means the reciprocal
2. 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
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 116: the fourth root of the top and of the bottom: 1 and 2, so it is 12.
4. 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. a^1/2 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. Write both sides as powers of the same base, then compare the indices.

1. 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
2. Replace them in the equation. Use the power-of-a-power law: (a^m)ⁿ = a^m×n. So raising 2² to the power x multiplies the indices.
   (2²)^x = 2^2x, so 2^2x = 2³
3. Equate the indices. The bases are the same, so the indices must be equal.
   2x = 3
4. Solve. Divide both sides by 2.
   x = 32 = 32

Check. Put x = 32 back: 4^3/2 = (2²)^3/2 = 2^{2 × (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. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.

Method. Simplify √12 first so the two surds are like surds, add them, then square.

1. Simplify √12. The largest square factor of 12 is 4.
   √12 = √(4 × 3) = 2√3
2. Add the like surds. √3 + 2√3 = 3√3 (1 lot plus 2 lots).
   √3 + √12 = √3 + 2√3 = 3√3
3. Square the result. Square the 3 and square the √3.
   (3√3)² = 3² × (√3)² = 9 × 3
4. Write the conclusion.
   9 × 3 = 27

Check. Decimals: 1.732 + 3.464 = 5.196 and 5.196² = 27.0. ✓

Watch out. Marks: 1 for simplifying √12, 1 for adding, 1 for squaring, 1 for the conclusion.