Question
Simplify $\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}+\sqrt{11}}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$
✓
Answer
$\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}+\sqrt{11}}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$
$=\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}+\sqrt{11}}\times\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}-\sqrt{11}}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}\times\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}+\sqrt{11}}$
$=\frac{\big(\sqrt{13}-\sqrt{11}\big)^2}{\big(\sqrt{13}\big)^2-\big(\sqrt{11}\big)^2}+\frac{\big(\sqrt{13}+\sqrt{11}\big)^2}{\big(\sqrt{13}\big)^2-\big(\sqrt{11}\big)^2}$
$=\frac{13+11-2\times\sqrt{13}\times\sqrt{11}}{13-11}+\frac{13+11+2\times\sqrt{13}\times\sqrt{11}}{13-11}$
$=\frac{24-2\sqrt{143}}{2}+\frac{24+2\sqrt{143}}{2}$
$=\frac{24-2\sqrt{143}+24+2\sqrt{143}}{2}$
$=\frac{48}{2}$
$=24$
$=\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}+\sqrt{11}}\times\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}-\sqrt{11}}+\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}\times\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}+\sqrt{11}}$
$=\frac{\big(\sqrt{13}-\sqrt{11}\big)^2}{\big(\sqrt{13}\big)^2-\big(\sqrt{11}\big)^2}+\frac{\big(\sqrt{13}+\sqrt{11}\big)^2}{\big(\sqrt{13}\big)^2-\big(\sqrt{11}\big)^2}$
$=\frac{13+11-2\times\sqrt{13}\times\sqrt{11}}{13-11}+\frac{13+11+2\times\sqrt{13}\times\sqrt{11}}{13-11}$
$=\frac{24-2\sqrt{143}}{2}+\frac{24+2\sqrt{143}}{2}$
$=\frac{24-2\sqrt{143}+24+2\sqrt{143}}{2}$
$=\frac{48}{2}$
$=24$
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