1.10 Fractional indices
9 exam questions worth 18 marks. Write your answer, open the mark scheme, then give yourself the marks you earned. Your answers are saved on this device.
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What is the value of 49^{\frac{1}{2}}?
Choose one answer.
Correct Not quite: the answer is 7
Mark scheme
B: 7
Model answer
7 (B). A power of 1/2 means the square root.
Work out (a) 64^{\frac{1}{3}} (b) 81^{\frac{1}{4}}.
Mark scheme
B1: 4; B1: 3
Model answer
(a) cube root of 64 = 4. (b) fourth root of 81 = 3.
Work out 8^{\frac{2}{3}}.
Mark scheme
M1: cube root of 8 = 2; A1: 4
Model answer
8^(2/3) = (∛8)² = 2² = 4.
Work out 16^{\frac{3}{4}}.
Mark scheme
M1: fourth root of 16 = 2; A1: 8
Model answer
16^(3/4) = (⁴√16)³ = 2³ = 8.
Work out 27^{-\frac{2}{3}}. Give your answer as a fraction.
Mark scheme
M1: 27^(2/3) = 9 (or 3²); A1: 1/9
Model answer
27^(2/3) = 3² = 9, and the negative index gives the reciprocal: 1/9.
Work out \left(\dfrac{4}{9}\right)^{-\frac{1}{2}}.
Mark scheme
M1: turns the fraction over (9/4) or finds √(4/9) = 2/3; A1: 3/2
Model answer
Turn the fraction upside down: (9/4)^(1/2) = √9 / √4 = 3/2.
Simplify (a) x^{\frac{1}{2}} \times x^{\frac{3}{2}} (b) \left(x^{\frac{2}{3}}\right)^6.
Mark scheme
B1: x²; B1: x⁴
Model answer
(a) Add the indices: 1/2 + 3/2 = 2, so x². (b) Multiply: (2/3) × 6 = 4, so x⁴.
Simplify (8x^6)^{\frac{1}{3}}.
Mark scheme
B1: 8^(1/3) = 2; B1: 2x²
Model answer
8^(1/3) = 2 and (x⁶)^(1/3) = x², so the answer is 2x².
Solve 4^x = 8.
Mark scheme
M1: writes both as powers of 2 (2^(2x) = 2³); M1: 2x = 3; A1: x = 3/2
Model answer
4 = 2² so 4x = 2^(2x). 8 = 2³. So 2x = 3 and x = 3/2.
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