Answer
$\displaystyle \frac{13\sqrt{11}+39}{2}$
Work Step by Step
We lose the square roots in the denominator by applying the difference of squares formula:
$(a\sqrt{x}+b\sqrt{y})(a\sqrt{x}-b\sqrt{y})=(a\sqrt{x})^{2}-(b\sqrt{y})^{2}=a^{2}x-b^{2}y$
$\displaystyle \frac{13}{\sqrt{11}-3} \displaystyle \color{red}{ \cdot\frac{\sqrt{11}+3}{\sqrt{11}+3} }\qquad$ (rationalize)
$ =\displaystyle \frac{13(\sqrt{11}+3)}{(\sqrt{11})^{2}-3^{2}} =\frac{13(\sqrt{11}+31)}{11-9} =\frac{13(\sqrt{11}+3)}{2}$
= $\displaystyle \frac{13\sqrt{11}+39}{2}$
