Printable · GCSE Higher · ages 14-16
Calculating with roots and indices worksheet — GCSE Higher
Fifteen questions on "calculating with roots and indices" — DfE statement N7. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
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Calculating with roots and indices worksheet — GCSE Higher
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- 1.Simplify (5² × 2³)³ ÷ (5³ × 2²)²
- 2.Work out the value of 2⁻⁴
- 3.Simplify 2³ × 2⁴, giving your answer as a single power of 2.
- 4.Decide which of 2³⁰ and 3²⁰ is the larger number, and write down the correct statement.
- 5.Simplify x⁶ ÷ x²
- 6.Work out the value of (√5)⁴
- 7.Write ∛(x²) as a single power of x.
- 8.Simplify x⁷ × x⁴, giving your answer as a single power of x.
- 9.A square petri dish has an area of 121 mm². A scientist wants to know its side length. Work out the side length of the dish.
- 10.Simplify x⁽³⁄⁴⁾ ÷ x⁽¹⁄⁴⁾
- 11.Work out the value of (1/3)⁻²
- 12.Work out the value of .
- 13.Simplify (x⁻²)³, giving your answer as a fraction.
- 14.Simplify (3a²)² × (2a)³
- 15.Simplify x⁵ × x³ ÷ x², giving your answer as a single power of x.
Answer key
- (b) 2⁵ — Method: a power outside brackets multiplies the index of every factor inside them, and dividing powers of the same base subtracts their indices. Working: (5² × 2³)³ = 5⁶ × 2⁹ and (5³ × 2²)² = 5⁶ × 2⁴. Dividing gives 5⁶⁻⁶ × 2⁹⁻⁴, and since 5⁰ = 1 the whole expression reduces to 2⁵. Answer: 2⁵. The distractors: 2² comes from adding the outside index to each inside index instead of multiplying, which gives 5⁵ × 2⁶ over 5⁵ × 2⁴; 2¹³ comes from adding the indices of 2 when dividing, 9 + 4, instead of subtracting them; 2 comes from applying the outside power to the first factor inside each bracket only, leaving 5⁶ × 2³ over 5⁶ × 2².
- (d) 1/16 — Method: a negative index means the reciprocal of the power, so 2⁻⁴ is 1 divided by 2⁴. Working: 2⁴ = 2 × 2 × 2 × 2 = 16, so the value is 1/16. Answer: 1/16. The distractors: −16 comes from reading the negative index as a minus sign on the result; 16 comes from ignoring the minus sign and working out 2⁴; 1/8 comes from multiplying the base by the index, 2 × 4, and writing 1 over that product.
- (d) 2⁷ — When multiplying powers of the same base, the indices add: 3 + 4 = 7, so 2³ × 2⁴ = 2⁷. Multiplying the indices instead of adding them gives 3 × 4 = 12, so 2¹². Subtracting the indices instead of adding them gives 4 − 3 = 1, so 2¹. Multiplying the bases together as well as adding the indices gives 2 × 2 = 4, so 4⁷.
- (b) 3²⁰ is larger — Method: two powers with different bases and different indices can be compared once they are rewritten with a common index, which is possible whenever the indices share a factor. Working: 30 and 20 have a highest common factor of 10, so 2³⁰ = (2³)¹⁰ = 8¹⁰ and 3²⁰ = (3²)¹⁰ = 9¹⁰. Both are now tenth powers, and since 9 is larger than 8, 9¹⁰ is larger than 8¹⁰. Answer: 3²⁰ is larger. The distractors: 2³⁰ is larger comes from comparing only the indices and choosing the power with the bigger index; They are equal comes from multiplying base by index, 2 × 30 and 3 × 20, and finding 60 each time; They cannot be compared without a calculator comes from assuming that powers this large can only be ranked by evaluating them in full.
- (d) x⁴ — Method: dividing two powers of the same letter subtracts the index of the divisor from the index of the term being divided. Working: six factors of x on the top and two on the bottom cancel in pairs, leaving 6 − 2 = 4 factors of x. Answer: x⁴. The distractors: x³ comes from dividing the indices, 6 ÷ 2, instead of subtracting them; x⁸ comes from adding the indices, 6 + 2, as though the powers were being multiplied; x¹² comes from multiplying the indices, 6 × 2, as though a power were being raised to a power.
- (a) 25 — Method: squaring a square root removes the root, so a square root raised to the power 4 can be squared in two stages. Working: (√5)⁴ = ((√5)²)² = 5² = 5 × 5 = 25. Answer: 25. The distractors: 5 comes from squaring once and stopping, treating the fourth power as a square; 625 comes from raising 5 to the power 4 and ignoring the root sign altogether; 20 comes from multiplying 5 by the index 4 instead of raising 5 to that power.
- (c) x⁽²⁄³⁾ — Method: a root can be written as a fractional index, with the root's index as the denominator and the power inside the root as the numerator. Working: the cube root gives a denominator of 3 and the square inside gives a numerator of 2, so ∛(x²) = x⁽²⁄³⁾. Answer: x⁽²⁄³⁾. The distractors: x⁽³⁄²⁾ comes from writing the fraction upside down, with the root's index on top; x⁽¹⁄⁶⁾ comes from treating the square as a second root and multiplying 1/3 by 1/2; x⁶ comes from multiplying the root's index by the power, 3 × 2, and keeping the result as a whole-number index.
- (a) x¹¹ — Method: when multiplying powers of the same base, add the indices. Working: 7 + 4 = 11, so x⁷ × x⁴ = x¹¹. x²⁸ comes from multiplying the indices, 7 × 4 = 28, instead of adding them. x³ comes from working out 7 − 4 = 3, which is the rule for dividing powers, not multiplying them. 11x comes from adding the indices to make 11 but then treating x as a coefficient instead of a power. Answer: x¹¹.
- (d) 11 mm — Method: for a square, the side length is the square root of the area. Working: 11 × 11 = 121, so the side length is 11 mm. 60.5 mm comes from working out 121 ÷ 2 = 60.5, halving the area instead of finding its square root. 242 mm comes from working out 121 × 2 = 242, doubling the area instead of finding its square root. 22 mm comes from working out 11 × 2 = 22, doubling the correct side length. Answer: 11 mm.
- (d) x⁽¹⁄²⁾ — Method: dividing two powers of the same letter subtracts the index of the divisor from the index of the term being divided, and fractional indices are subtracted like any other fractions. Working: 3/4 − 1/4 = 2/4, which simplifies to 1/2, so the result is x⁽¹⁄²⁾. Answer: x⁽¹⁄²⁾. The distractors: x comes from adding the indices, 3/4 + 1/4 = 1, as though the powers were being multiplied; x³ comes from dividing the indices, so that 3/4 divided by 1/4 gives 3; x⁽³⁄¹⁶⁾ comes from multiplying the indices, 3/4 × 1/4.
- (b) 9 — Method: a negative index means take the reciprocal of the base first and then apply the positive power. Working: the reciprocal of 1/3 is 3, so (1/3)⁻² = 3² = 3 × 3 = 9. Answer: 9. The distractors: 1/9 comes from ignoring the minus sign and squaring 1/3 as it stands; −9 comes from reading the negative index as a minus sign on the result; 6 comes from multiplying the denominator by the index, 3 × 2, instead of squaring the reciprocal.
- (a) 17 — Method: work out each power separately, then combine them as the question asks. Working: $2^3 = 8$ and $3^2 = 9$, and 8 + 9 = 17. 72 comes from working out 8 × 9 = 72, multiplying the two powers instead of adding them. 12 comes from misreading the powers as repeated multiplication of the base by the index, 2 × 3 + 3 × 2 = 6 + 6 = 12. −1 comes from working out 8 − 9 = −1, subtracting the powers instead of adding them. Answer: 17.
- (c) 1/x⁶ — Method: raising a power to another power multiplies the two indices, and a negative index means one over the matching positive power. Working: −2 × 3 = −6, so (x⁻²)³ = x⁻⁶, and x⁻⁶ written as a fraction is 1/x⁶. Answer: 1/x⁶. The distractors: x⁶ comes from multiplying the indices correctly but dropping the minus sign; 1/x⁵ comes from adding the sizes of the indices, 2 + 3, instead of multiplying them; −x⁶ comes from reading the negative index as a minus sign in front of the whole term.
- (d) 72a⁷ — Method: a power outside brackets applies to every factor inside them, and multiplying two powers of the same letter adds their indices. Working: (3a²)² = 3² × a⁴ = 9a⁴, and (2a)³ = 2³ × a³ = 8a³. Multiplying the two results gives 9 × 8 = 72 for the number and 4 + 3 = 7 for the index of a. Answer: 72a⁷. The distractors: 36a⁷ comes from squaring the 2 in (2a)³ instead of cubing it, giving 4a³ and then 9 × 4; 72a¹² comes from multiplying the indices 4 and 3 when the two terms are multiplied, instead of adding them; 17a⁷ comes from adding the coefficients 9 and 8 rather than multiplying them.
- (a) x⁶ — Method: work through the powers in order — multiplying powers of the same base means adding indices, and dividing powers of the same base means subtracting indices. Working: first, x⁵ × x³ = x⁸ (adding 5 and 3); then x⁸ ÷ x² = x⁶ (subtracting 2 from 8). x⁴ comes from swapping the two rules — subtracting for the multiplication, 5 − 3 = 2, and then adding for the division, 2 + 2 = 4. x¹⁰ comes from adding all three indices, 5 + 3 + 2 = 10, treating the division the same as a multiplication. 6x comes from correctly reaching a total index of 6 but then writing it as a coefficient of x instead of as its power. Answer: x⁶.
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