# Surds

## Simplifying Surds

The trick to simplifying surds is to consider the number within the square root and see if you can identify any square factors of this number. Square factors are numbers that you can divide by (and obtain an integer result) that happen to be square numbers: 1,4,9,16,25,…

Example 1: $\sqrt{8}=\sqrt{4\times 2}=\sqrt{4}\times \sqrt{2}=2\sqrt{2}$

Example 2: $\sqrt{27}=\sqrt{9\times 3}=\sqrt{9}\times \sqrt{3}=3\sqrt{3}$

Example 3: $\sqrt{20}=\sqrt{4\times 5}=\sqrt{4}\times \sqrt{5}=2\sqrt{5}$

Example 4: $\sqrt{32}=\sqrt{16\times 2}=\sqrt{16}\times \sqrt{2}=4\sqrt{2}$

Example 5: $\sqrt{75}=\sqrt{25\times 3}=\sqrt{25}\times \sqrt{3}=5\sqrt{3}$

Example 6: $\sqrt{98}=\sqrt{49\times 2}=\sqrt{49}\times \sqrt{2}=7\sqrt{2}$

## Manipulating Surds

When manipulating surds or simplifying algebraic expressions involving surds it is useful to remember, in addition to how to simplify surds, the following:
$\sqrt{x}\times \sqrt{x}=x$
$x\times \sqrt{y}=\sqrt{y}\times x=x\sqrt{y}$

Example 1: $\left(\sqrt{3}+2\right)^2=\left(\sqrt{3}+2\right)\left(\sqrt{3}+2\right)=\sqrt{3}\times\sqrt{3}+2\times\sqrt{3}+\sqrt{3}\times 2 +2\times 2=3+2\sqrt{3}+2\sqrt{3}+4=7+4\sqrt{3}$

Example 2: $(7-\sqrt{5})(4+3\sqrt{5})=28-4\sqrt{5}+21\sqrt{5}-5=23+17\sqrt{3}$

## Rationalising the Denominator

Fractions can have a surd in the denominator and you may be required to rationalise the denominator. The idea is to find an equivalent fraction, by multiplying or dividing the top and bottom by the same number, where the surd has been eliminated from the denominator. The surd can appear in the numerator, however.

Example 1 – $\frac{5}{\sqrt{3}}=\frac{5}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{5\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{5\sqrt{3}}{9}$
Example 2 – $\frac{6}{\sqrt{8}}=\frac{6}{2\sqrt{2}}=\frac{3}{\sqrt{2}}=\frac{3}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{3\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{3\sqrt{2}}{2}$