# Rationalizing quotients [duplicate]

I have

$$\frac{\sqrt{10}}{\sqrt{5} - 2}$$

I have no idea what to do, I know that I can do some tricks with splitting square roots up but pulling out whole numbers like I know that $\sqrt{27}$ is just $\sqrt{3*9}$ so I can pull out a nine which becomes a 3. Here though I have no such options, what can I possibly do?

## 4 Answers

Hint

$$\frac{1}{\sqrt{5}-2}=\frac{\sqrt{5}+2}{(\sqrt{5}+2)}\frac{1}{(\sqrt{5}-2)}$$

and $(a+b)(a-b)=a^2-b^2$

• Oh I got it, I forgot that trick. Thanks. – user138246 May 18 '13 at 18:45

$$\frac{\sqrt{10}}{\sqrt{5} - 2}=\frac{\sqrt{10}}{\sqrt{5} - 2}\cdot\frac{{\sqrt{5} + 2}}{{\sqrt{5} + 2}}=\frac{10({\sqrt{5}+2})}{5-4}=10({\sqrt{5}+2})$$

HINT: Multiply the fraction by $1$ in the cleverly chosen disguise of

$$\frac{\sqrt5+2}{\sqrt5+2}\;.$$

Then recognize the resulting denominator as a familiar factorization of something nice.

If all you need to do is rationalize the denominator, try multiplying the numerator and denominator by $\sqrt{5}+2$.