Is \sqrt{a+b}=\sqrt a+\sqrt b?

No. For example \sqrt{9+16}=5 but \sqrt9+\sqrt{16}=7.

What does it mean to rationalise a denominator?

Rewrite the fraction so that the denominator has no surd.

How do you rationalise \dfrac{1}{\sqrt a}?

Multiply top and bottom by \sqrt a: \dfrac{\sqrt a}{a}.

What do you multiply by to rationalise \dfrac{1}{a+\sqrt b}?

The conjugate a-\sqrt b (top and bottom). The new denominator is a^2-b.

Rationalise \dfrac{2}{3-\sqrt5}.

\dfrac{2(3+\sqrt5)}{4}=\dfrac{3+\sqrt5}{2}

SurdsAQA A-Level Maths: Flashcards

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What is a surd?

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What is a surd?: An irrational root, such as $\sqrt2$ , left in root form to keep an answer exact.
Simplify $\sqrt{ab}$ .: $\sqrt a\times\sqrt b$
What is $\sqrt a\times\sqrt a$ ?: $a$
Simplify $\sqrt{50}$ .: $5\sqrt2$
Simplify $\sqrt{72}$ .: $6\sqrt2$ (largest square factor is 36).
Is $\sqrt{a+b}=\sqrt a+\sqrt b$ ?: No. For example $\sqrt{9+16}=5$ but $\sqrt9+\sqrt{16}=7$ .
Expand $(3+\sqrt5)^2$ .: $9+6\sqrt5+5=14+6\sqrt5$
Expand $(a+\sqrt b)(a-\sqrt b)$ .: $a^2-b$ , which is rational.
What does it mean to rationalise a denominator?: Rewrite the fraction so that the denominator has no surd.
How do you rationalise $\dfrac{1}{\sqrt a}$ ?: Multiply top and bottom by $\sqrt a$ : $\dfrac{\sqrt a}{a}$ .
What do you multiply by to rationalise $\dfrac{1}{a+\sqrt b}$ ?: The conjugate $a-\sqrt b$ (top and bottom). The new denominator is $a^2-b$ .
Rationalise $\dfrac{2}{3-\sqrt5}$ .: $\dfrac{2(3+\sqrt5)}{4}=\dfrac{3+\sqrt5}{2}$
Simplify $\dfrac{5\sqrt2}{2\sqrt2}$ .: $\dfrac52$

Exam questions on Surds

Let $p=\sqrt{50}$ and $q=\sqrt{8}$ .
Express $\dfrac{p}{q}$ as a fraction in its simplest form.2 marks
A rectangle has length $(3+\sqrt5)$ cm and width $(3-\sqrt5)$ cm.
Express length divided by width in the form $a+b\sqrt5$ , where $a$ and $b$ are rational.2 marks
A rectangle has area $(8+2\sqrt7)$ cm $^2$ and length $(\sqrt7-1)$ cm.
Find the width of the rectangle, giving your answer in the form $\dfrac{a+b\sqrt7}{c}$ where $a$ , $b$ and $c$ are integers.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).