Answer
$=8\sqrt{5}-16$
Work Step by Step
We lose the square roots in the denominator by applying the difference of squares formula:
$(a\sqrt{x}+b\sqrt{y})(a\sqrt{x}-b\sqrt{y})=(a\sqrt{x})^{2}-(b\sqrt{y})^{2}=a^{2}x-b^{2}y$
$\displaystyle \frac{8}{\sqrt{5}+2} \displaystyle \color{red}{ \cdot\frac{\sqrt{5}-2}{\sqrt{5}-2} }\qquad$ (rationalize)
$ =\displaystyle \frac{8(\sqrt{5}-2)}{(\sqrt{5})^{2}-2^{2}} =\frac{8(\sqrt{5}-2)}{5-4} $
$=8(\sqrt{5}-2)$
$=8\sqrt{5}-16$
