Question
Simplify:
$\frac{1}{\sqrt{3}+\sqrt{2}}-\frac{2}{\sqrt{5}-\sqrt{3}}-\frac{3}{\sqrt{2}-\sqrt{5}}$
$\frac{1}{\sqrt{3}+\sqrt{2}}-\frac{2}{\sqrt{5}-\sqrt{3}}-\frac{3}{\sqrt{2}-\sqrt{5}}$
✓
Answer
$\frac{1}{\sqrt{3}+\sqrt{2}}-\frac{2}{\sqrt{5}-\sqrt{3}}-\frac{3}{\sqrt{2}-\sqrt{5}}$
$=\frac{1}{\sqrt{3}-\sqrt{2}}\times\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}-\frac{2}{\sqrt{5}-\sqrt{3}}$ $\times\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}-\frac{3}{\sqrt{2}-\sqrt{5}}\times\frac{\sqrt{2}+\sqrt{5}}{\sqrt{2}+\sqrt{5}}$
$=\frac{\sqrt{3}+\sqrt{2}}{\big(\sqrt{3}\big)^2-\big(\sqrt{2}\big)^2}-\frac{2\big(\sqrt{5}+\sqrt{3}\big)}{\big(\sqrt{5}\big)^2-\big(\sqrt{3}\big)^2}-\frac{3\big(\sqrt{2}+\sqrt{5}\big)}{\big(\sqrt{2}\big)^2-\big(\sqrt{5}\big)^2}$
$=\frac{\sqrt{3}+\sqrt{2}}{3-2}-\frac{2\big(\sqrt{5}+\sqrt{3}\big)}{5-3}-\frac{3\big(\sqrt{2}+\sqrt{5}\big)}{2-5}$
$=\frac{\sqrt{3}+\sqrt{2}}{1}-\frac{2\big(\sqrt{5}+\sqrt{3}\big)}{2}-\frac{3\big(\sqrt{2}+\sqrt{5}\big)}{-3}$
$=\big(\sqrt{3}+\sqrt{2}\big)-\big(\sqrt{5}+\sqrt{3}\big)+\big(\sqrt{2}+\sqrt{5}\big)$
$=\sqrt{3}+\sqrt{2}-\sqrt{5}-\sqrt{3}+\sqrt{2}+\sqrt{5}$
$=2\sqrt{2}$
$=\frac{1}{\sqrt{3}-\sqrt{2}}\times\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}-\frac{2}{\sqrt{5}-\sqrt{3}}$ $\times\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}-\frac{3}{\sqrt{2}-\sqrt{5}}\times\frac{\sqrt{2}+\sqrt{5}}{\sqrt{2}+\sqrt{5}}$
$=\frac{\sqrt{3}+\sqrt{2}}{\big(\sqrt{3}\big)^2-\big(\sqrt{2}\big)^2}-\frac{2\big(\sqrt{5}+\sqrt{3}\big)}{\big(\sqrt{5}\big)^2-\big(\sqrt{3}\big)^2}-\frac{3\big(\sqrt{2}+\sqrt{5}\big)}{\big(\sqrt{2}\big)^2-\big(\sqrt{5}\big)^2}$
$=\frac{\sqrt{3}+\sqrt{2}}{3-2}-\frac{2\big(\sqrt{5}+\sqrt{3}\big)}{5-3}-\frac{3\big(\sqrt{2}+\sqrt{5}\big)}{2-5}$
$=\frac{\sqrt{3}+\sqrt{2}}{1}-\frac{2\big(\sqrt{5}+\sqrt{3}\big)}{2}-\frac{3\big(\sqrt{2}+\sqrt{5}\big)}{-3}$
$=\big(\sqrt{3}+\sqrt{2}\big)-\big(\sqrt{5}+\sqrt{3}\big)+\big(\sqrt{2}+\sqrt{5}\big)$
$=\sqrt{3}+\sqrt{2}-\sqrt{5}-\sqrt{3}+\sqrt{2}+\sqrt{5}$
$=2\sqrt{2}$
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