<s>[INST] Rationalizing the denominator of $\frac {\sqrt{10}}{\sqrt{5} -2}$ I have the expression
$$\frac {\sqrt{10}}{\sqrt{5} -2}$$
I can't figure out what to do from here, I can't seem to pull any numbers out of either of the square roots so it appears that it must remain as is.
 [/INST] To rationalize the denominator i. e. turn the denominator rational multiply both numerator and denominator by
the conjugate of the denominator $-\sqrt{5}-2$ or its symmetric  $\sqrt{5}+2$, expand both and
simplify
$$\begin{eqnarray*}
\frac{\sqrt{10}}{\sqrt{5}-2} &=&\frac{\sqrt{10}\left( \sqrt{5}+2\right) }{
\left( \sqrt{5}-2\right) \left( \sqrt{5}+2\right) }=\frac{\sqrt{10}\sqrt{5}+\sqrt{10}\times 2}{\left( \sqrt{5}\right) ^{2}-2^{2}} \\
&=&\frac{\sqrt{50}+2\sqrt{10}}{5-4}=\frac{\sqrt{50}+2\sqrt{10}}{1}=\sqrt{50}%
+2\sqrt{10}.
\end{eqnarray*}.
$$
In general [Edited to correct] $$\frac{1}{a+\sqrt{b}}=\frac{a-\sqrt{b}}{\left( a+\sqrt{b}\right)
\left( a-\sqrt{b}\right) }=\frac{a-\sqrt{b}}{a^{2}-b}.$$
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