Higher June 2025 Paper 1 Q19
19
(a) Work out the value of \(\quad \left(\dfrac{9}{16}\right)^{-\frac{3}{2}}\) [3 marks]
(b) \(\sqrt{125} = 5^n\)
Work out the value of \(n\). [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{64}{27}\) or \(2\dfrac{10}{27}\) | B3 | B2 shows understanding of at least two of • a negative index means reciprocal • an index of \(\dfrac{1}{2}\) means square root • an index of 3 means cube B1 shows understanding of one of the above |
Additional guidance
| Condone missing brackets | |
| Allow equivalent decimals, fractions or mixed numbers throughout | |
| Ignore any attempt to simplify, change to a mixed number or convert to a decimal after a correct answer is seen eg \(\dfrac{64}{27} = 2\dfrac{20}{27}\) | B3 |
| Understanding can be shown in expressions which are not equivalent to the original eg \(\dfrac{27}{64}\) implies understanding of square root and cubing, but not reciprocal | B2 |
| Example B2 expressions: \(\left(\dfrac{1/\sqrt{9}}{1/\sqrt{16}}\right)^3\) or \(\left(\dfrac{4096}{729}\right)^{\frac{1}{2}}\) or \(\left(\dfrac{27}{64}\right)^{-1}\) | B2 |
| Example B1 expressions: \(\left(\dfrac{16}{9}\right)\) or \(\left(\dfrac{3}{4}\right)^{-3}\) or \(\left(\dfrac{729}{4096}\right)^{-\frac{1}{2}}\) | B1 |
| To show understanding, correct method or value for calculation of a cube is required, but calculation of a square root is not required eg1 \(\left(\dfrac{\sqrt{9}}{\sqrt{16}}\right)^3\) only shows understanding of the square root | B1 |
| eg2 \(\left(\dfrac{\sqrt{9}}{\sqrt{16}}\right) \times \left(\dfrac{\sqrt{9}}{\sqrt{16}}\right) \times \left(\dfrac{\sqrt{9}}{\sqrt{16}}\right)\) shows understanding of the square root and cubing | B2 |
| Answer | Mark | Comments |
|---|---|---|
| 1.5 or \(\dfrac{3}{2}\) or \(1\dfrac{1}{2}\) | B2 | B1 \(125^{\frac{1}{2}}\) or \(\left(5^3\right)^{\frac{1}{2}}\) or \(\left(5^{\frac{1}{2}}\right)^3\) or \(5^{\frac{3}{2}}\) or \(125 = 5^{2n}\) or \(5^3 = 5^{2n}\) or \(2n = 3\) |