Answer
$-10\sqrt{2}-5\sqrt{10}$
Work Step by Step
The given expression is
$=\frac{10}{\sqrt{8}-\sqrt{10}}$
The conjugate of $\sqrt{8}-\sqrt{10}$ is $\sqrt{8}+\sqrt{10}$
$=\frac{10}{\sqrt{8}-\sqrt{10}} \cdot \frac{\sqrt{8}+\sqrt{10}}{\sqrt{8}+\sqrt{10}}$
Use sum and difference pattern.
$=\frac{10(\sqrt{8}+\sqrt{10})}{(\sqrt{8})^2-(\sqrt{10})^2}$
Simplify.
$=\frac{10(\sqrt{8}+\sqrt{10})}{8-10}$
$=\frac{10(\sqrt{8}+\sqrt{10})}{-2}$
$=-5(\sqrt{8}+\sqrt{10})$
Use distributive property.
$=-5\sqrt{8}-5\sqrt{10}$
Factor as square terms.
$=-5\sqrt{4\cdot 2}-5\sqrt{10}$
Use product property of square roots.
$=-5\sqrt{4}\cdot \sqrt{2}-5\sqrt{10}$
Simplify.
$=-5(2)\sqrt{2}-5\sqrt{10}$
$=-10\sqrt{2}-5\sqrt{10}$
