conjugate surds

short intro to using conjugate surds

Say we have $\displaystyle \frac{1}{\sqrt{13}-3}$ and need the $\sqrt{13}$ on the top instead.

This is done by multiplying the top and bottom by the ‘conjugate surd’ – which just means changing the + or – sign.

So multiplying by $\displaystyle \frac{\sqrt{13}+3}{\sqrt{13}+3}$ , we get $\displaystyle \frac{\sqrt{13}+3}{(\sqrt{13}+3)(\sqrt{13}-3)}$

The point of doing this is that the bottom becomes

$\sqrt{13}\times\sqrt{13}+3\sqrt{13}-3\sqrt{13}-9 = 13 - 9 = 4$

So the transformed fraction is $\displaystyle\frac{\sqrt{13}+3}{4}$ .

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