Answer
$\frac{2\sqrt{21}-2\sqrt{6}}{5}$.
Work Step by Step
The given expression is
$=\frac{\sqrt{12}}{\sqrt{2}+\sqrt{7}}$
$=\frac{\sqrt{12}}{\sqrt{7}+\sqrt{2}}$
The conjugate of $\sqrt{7}+\sqrt{2}$ is $\sqrt{7}-\sqrt{2}$.
Multiply by $\frac{\sqrt{7}-\sqrt{2}}{\sqrt{7}-\sqrt{2}}$.
$=\frac{\sqrt{12}}{\sqrt{7}+\sqrt{2}}\cdot \frac{\sqrt{7}-\sqrt{2}}{\sqrt{7}-\sqrt{2}}$
Use sum and difference pattern.
$=\frac{\sqrt{12}(\sqrt{7}-\sqrt{2})}{(\sqrt{7})^2-(\sqrt{2})^2}$
Simplify.
$=\frac{\sqrt{12}(\sqrt{7}-\sqrt{2})}{7-2}$
$=\frac{\sqrt{12}(\sqrt{7}-\sqrt{2})}{5}$
Use $\sqrt{12}=2\sqrt{3}$.
$=\frac{2\sqrt{3}(\sqrt{7}-\sqrt{2})}{5}$
Use distributive property.
$=\frac{2\sqrt{3}\cdot \sqrt{7}-2\sqrt{3}\cdot \sqrt{2})}{5}$
Use product property of square roots.
$=\frac{2\sqrt{3\cdot 7}-2\sqrt{3\cdot 2}}{5}$
Simplify.
$=\frac{2\sqrt{21}-2\sqrt{6}}{5}$.
