Surds, exact calculations and rationalising

3 exam-style questions. Answers and the mark for each step are on the last page.

Question 1Non-calculator · 5 marks
Do not use a calculator. Give each answer in its simplest exact form.
(a) Simplify $\sqrt{48}$ (1)
(b) Expand and simplify $(3 + \sqrt{2})(3 - \sqrt{2})$ (2)
(c) Rationalise the denominator of $\dfrac{6}{\sqrt{3}}$ and simplify. (2)
Question 2Non-calculator · 4 marks
(a) Simplify 1/( $\sqrt{7}$ + $\sqrt{3}$ ) + 1/( $\sqrt{7}$ $-$ $\sqrt{3}$ ). Give your answer with a rational denominator. (4)
Question 3Non-calculator · 6 marks
A rectangle has length $(5 + \sqrt{3})$ cm and area $(17 - \sqrt{3})\ \text{cm}^2$ .
(a) Find the width of the rectangle. Give your answer in the form $a + b\sqrt{3}$ , where $a$ and $b$ are integers. (4)
(b) Show that the perimeter of the rectangle is 18 cm. (2)

Answers and marks

(a) $4\sqrt{3}$

(b) $7$

M1 Four terms with at least three correct, or using the difference of two squares $9 - 2$ .
A1 The correct answer, $7$ .

(c) $2\sqrt{3}$

(a) $\sqrt{7}/2$

M1 Establishing $\frac{\sqrt{7}-\sqrt{3}}{4}$ or an equivalent valid method.
M1 Establishing $\frac{\sqrt{7}+\sqrt{3}}{4}$ or an equivalent valid method.
M1 Establishing $(\sqrt{7}-\sqrt{3})/4+(\sqrt{7}+\sqrt{3})/4$ or an equivalent valid method.
A1 Correct answer: $\sqrt{7}/2$

(a) $(4 - \sqrt{3})$ cm

(b) $2(5 + \sqrt{3}) + 2(4 - \sqrt{3}) = 18$

Surds, exact calculations and rationalising

Type your working online and see every mark you earned and lost.