A-level · Algebra & functions

Surds and indicesExam-style questions

Powers and roots, kept exact. The three index laws and why they hold; why a zero power is 1 and a negative power is "one over"; fractional powers as a root and a power; writing an expression as powers of x, ready for calculus; simplifying surds, and the root of a sum that does not split; and clearing a surd from a denominator with a conjugate.

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16 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4

1
Given that

$y = \dfrac{(2x + 3)^2}{\sqrt{x}}, \qquad x > 0$

find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ , giving each term in its simplest form. (3 marks)
Level 2 · 3 marksTry it
2
Given that

$p = 5\sqrt[3]{x} \qquad\qquad q = \frac{1}{5x^{3}}$

express each of the following in the form $kx^{n}$ where $k$ and $n$ are fully simplified constants.
(a) $q$

(1 mark)
(b) $\dfrac{p^{2}}{q}$

(2 marks)
(c) $\left(p^{3}q\right)^{-\frac{1}{2}}$

(2 marks)
Level 2 · 5 marksTry it
3
Express

$\frac{7 - \sqrt[3]{x^4}}{x^3}$

in the form $7x^p - x^q$ , where $p$ and $q$ are constants.

(2 marks)
Level 2 · 2 marksTry it