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Surds

A surd is a root that will not come out as a whole number, so we keep it exact rather than rounding. Two skills, simplifying and rationalising, do nearly all the work.

MYP 4Standard MathsNumberCriteria A · C~8 min read

Written by Fayek Jawad Haider · MYP student

What a surd is

A surd is a square root that cannot be simplified to a whole number, so we leave it in root form to keep it exact. \(\sqrt{2}\), \(\sqrt{3}\) and \(\sqrt{10}\) are surds; \(\sqrt{9}=3\) is not, because it tidies to a whole number.

Surd
A root whose exact value is irrational, so it is left written with the root sign, such as \(\sqrt{5}\).

One rule for multiplying roots does most of the heavy lifting:

\[ \sqrt{a} \times \sqrt{b} = \sqrt{ab} \]

You cannot add roots inside one sign

\(\sqrt{a}+\sqrt{b}\) is not \(\sqrt{a+b}\). For example \(\sqrt{9}+\sqrt{16}=3+4=7\), but \(\sqrt{25}=5\). You can only add surds that are already "like", such as \(2\sqrt{3}+4\sqrt{3}=6\sqrt{3}\).

Simplifying surds

To simplify a surd, split the number under the root into a factor pair where one factor is a perfect square (1, 4, 9, 16, 25, 36, and so on). Then take the square root of that factor outside the sign. Using the largest square factor gets you there in one step.

Worked example

Simplify \(\sqrt{72}\).

1
Find the largest perfect-square factor of 72: that is 36, since \(72 = 36 \times 2\).
2
Split the root using the multiplying rule: \(\sqrt{72} = \sqrt{36}\times\sqrt{2}\).
3
\(\sqrt{36}=6\), so it comes outside the root.
\(\sqrt{72} = 6\sqrt{2}\)

Rationalising the denominator

Convention says we do not leave a surd on the bottom of a fraction. Rationalising the denominator clears it, by multiplying the top and bottom by that same surd. Since you multiply top and bottom by the same thing, the value of the fraction does not change, it is really just multiplying by 1.

Worked example

Rationalise \(\dfrac{5}{\sqrt{5}}\).

1
Multiply top and bottom by \(\sqrt{5}\): \(\dfrac{5}{\sqrt{5}}\times\dfrac{\sqrt{5}}{\sqrt{5}} = \dfrac{5\sqrt{5}}{5}\).
2
The denominator is now the whole number 5 (since \(\sqrt{5}\times\sqrt{5}=5\)). Simplify the fraction: \(\dfrac{5}{5}=1\).
\(\dfrac{5}{\sqrt{5}} = \sqrt{5}\)

Where this is assessed

Surds sit in Criterion A (choosing the right square factor) and Criterion C (exact, correctly-notated answers). Exam questions often demand an exact surd form, so a rounded decimal such as \(6\sqrt{2}\approx 8.49\) would lose the marks.

Check yourself

1. Simplify \(\sqrt{48}\). +

Largest square factor of 48 is 16, since \(48 = 16 \times 3\). So \(\sqrt{48} = \sqrt{16}\times\sqrt{3} = \mathbf{4\sqrt{3}}\).

2. Simplify \(\sqrt{200}\). +

Largest square factor of 200 is 100, since \(200 = 100 \times 2\). So \(\sqrt{200} = \sqrt{100}\times\sqrt{2} = \mathbf{10\sqrt{2}}\).

3. Rationalise \(\dfrac{6}{\sqrt{2}}\). +

Multiply top and bottom by \(\sqrt{2}\): \(\dfrac{6}{\sqrt{2}}\times\dfrac{\sqrt{2}}{\sqrt{2}} = \dfrac{6\sqrt{2}}{2}\). Then \(\dfrac{6}{2}=3\), giving \(\mathbf{3\sqrt{2}}\).

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