Question
Simplify : $\frac{7 \sqrt{3}}{\sqrt{10}+\sqrt{3}}-\frac{2 \sqrt{5}}{\sqrt{6}+\sqrt{5}}-\frac{3 \sqrt{2}}{\sqrt{15}+3 \sqrt{2}}$.
✓
Answer
$\frac{7 \sqrt{3}}{\sqrt{10}+\sqrt{3}}-\frac{2 \sqrt{5}}{\sqrt{6}+\sqrt{5}}-\frac{3 \sqrt{2}}{\sqrt{15}+3 \sqrt{2}}$
$=\frac{7 \sqrt{3}}{\sqrt{10}+\sqrt{3}} \times \frac{\sqrt{10}-\sqrt{3}}{\sqrt{10}-\sqrt{3}}-\frac{2 \sqrt{5}}{\sqrt{6}+\sqrt{5}} \times \frac{\sqrt{6}-\sqrt{5}}{\sqrt{6}-\sqrt{5}}-\frac{3 \sqrt{2}}{\sqrt{15}+3 \sqrt{2}} \times \frac{\sqrt{15}-3 \sqrt{2}}{\sqrt{15}-3 \sqrt{2}}$
$=\frac{7 \sqrt{3}(\sqrt{10}-\sqrt{3})}{10-3}-\frac{2 \sqrt{5}(\sqrt{6}-\sqrt{5})}{6-5}-\frac{3 \sqrt{2}(\sqrt{15}-3 \sqrt{2})}{15-18}$
$=\sqrt{3}(\sqrt{10}-\sqrt{3})-2 \sqrt{5}(\sqrt{6}-\sqrt{5})+\sqrt{2}(\sqrt{15}-3 \sqrt{2})$
$=\sqrt{30}-3-2 \sqrt{30}+10+\sqrt{30}-6$
$=2 \sqrt{30}-9-2 \sqrt{30}+10$
$=1$
$=\frac{7 \sqrt{3}}{\sqrt{10}+\sqrt{3}} \times \frac{\sqrt{10}-\sqrt{3}}{\sqrt{10}-\sqrt{3}}-\frac{2 \sqrt{5}}{\sqrt{6}+\sqrt{5}} \times \frac{\sqrt{6}-\sqrt{5}}{\sqrt{6}-\sqrt{5}}-\frac{3 \sqrt{2}}{\sqrt{15}+3 \sqrt{2}} \times \frac{\sqrt{15}-3 \sqrt{2}}{\sqrt{15}-3 \sqrt{2}}$
$=\frac{7 \sqrt{3}(\sqrt{10}-\sqrt{3})}{10-3}-\frac{2 \sqrt{5}(\sqrt{6}-\sqrt{5})}{6-5}-\frac{3 \sqrt{2}(\sqrt{15}-3 \sqrt{2})}{15-18}$
$=\sqrt{3}(\sqrt{10}-\sqrt{3})-2 \sqrt{5}(\sqrt{6}-\sqrt{5})+\sqrt{2}(\sqrt{15}-3 \sqrt{2})$
$=\sqrt{30}-3-2 \sqrt{30}+10+\sqrt{30}-6$
$=2 \sqrt{30}-9-2 \sqrt{30}+10$
$=1$
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