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.

Start practising Read the lesson

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
Show that $\dfrac{1}{\sqrt{n+1} + \sqrt{n}} \equiv \sqrt{n+1} - \sqrt{n}$ for all $n \ge 0$ .
Level 3 · 3 marksTry it
2
Express $\dfrac{5}{2 - \sqrt{3}} - \dfrac{4}{2 + \sqrt{3}}$ in the form $a + b\sqrt{3}$ , where $a$ and $b$ are integers.
Level 3 · 3 marksTry it
3
Given that $5^{x} = 3$ , find the exact value of $5^{2x + 1}$ .
Level 3 · 2 marksTry it
4
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.
(a) Use the substitution $p = 5^{x}$ to show that the equation

$5^{2x+1} - 5^{x+3} = 5^{x} - 25$

can be rewritten as

$5p^{2} - 126p + 25 = 0$

(2 marks)
(b) Hence solve

$5^{2x+1} - 5^{x+3} = 5^{x} - 25$

(3 marks)
Level 3 · 5 marksTry it
5
Solve $2^{2x + 1} - 5\left(2^{x}\right) + 2 = 0$ .
Level 3 · 5 marksTry it

3 more questions on surds and indices

Free with an account — just an email, no card. The whole bank, every mark scheme and worked solution.

Start free