Answer
$3\sqrt{7}-3\sqrt{3}$
Work Step by Step
We lose the square roots in the denominator by applying the difference of squares formula:
$(a\sqrt{x}+b\sqrt{y})(a\sqrt{x}-b\sqrt{y})=(a\sqrt{x})^{2}-(b\sqrt{y})^{2}=a^{2}x-b^{2}y$
$\displaystyle \frac{12}{\sqrt{7}+\sqrt{3}} \displaystyle \color{red}{ \cdot\frac{\sqrt{7}-\sqrt{3}}{\sqrt{7}-\sqrt{3}} }\qquad$ (rationalize)
$ =\displaystyle \frac{12(\sqrt{7}-\sqrt{3})}{(\sqrt{7})^{2}-(\sqrt{3})^{2}} =\frac{12(\sqrt{7}-\sqrt{3})}{7-3} =\frac{12(\sqrt{7}-\sqrt{3})}{4}$
$=3(\sqrt{7}-\sqrt{3})$
= $3\sqrt{7}-3\sqrt{3}$
