Answer
$\displaystyle \frac{8+\sqrt{55}}{3}$
Work Step by Step
We lose the square roots in the denominator by applying the difference of squares formula:
$(a\sqrt{x}+b\sqrt{y})(a\sqrt{x}-b\sqrt{y})=(a\sqrt{x})^{2}-(b\sqrt{y})^{2}=a^{2}x-b^{2}y$
$\displaystyle \frac{\sqrt{11}-\sqrt{5}}{\sqrt{11}+\sqrt{5}}\color{red}{ \cdot\frac{\sqrt{11}-\sqrt{5}}{\sqrt{11}-\sqrt{5}} }\qquad$ (rationalize)
$=\displaystyle \frac{(\sqrt{11}-\sqrt{5})^{2}}{(\sqrt{11})^{2}-(\sqrt{5})^{2}}$
... the numerator is a square of a sum, $(A+B)^{2}=A^{2}+2AB+B^{2}$
$=\displaystyle \frac{(\sqrt{11})^{2}+2\sqrt{11}\cdot\sqrt{5}+(\sqrt{5})^{2}}{11-5}$
$=\displaystyle \frac{11+2\sqrt{55}+5}{6}$
$=\displaystyle \frac{16+2\sqrt{55}}{6}$
$=\displaystyle \frac{2(8+\sqrt{55})}{6}$
$=\displaystyle \frac{8+\sqrt{55}}{3}$
