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  • Simplify :- 7sqroot 3 by sqroot10+sqroot3-2sqroot5 by sqroot6+sqroot5-3sqroot2by sqroot15+3sqroot2

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}}

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Answer. 1
Solution. \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}}
Rationalise the denominators:
\Rightarrow \left ( \frac{7\sqrt{3}}{\sqrt{10}+\sqrt{3}}\times \frac{\sqrt{10}-\sqrt{3}}{\sqrt{10}-\sqrt{3}} \right )-\left ( \frac{2\sqrt{5}}{\sqrt{6}+\sqrt{3}} \times \frac{\sqrt{6}-\sqrt{5}}{\sqrt{6}-\sqrt{5}}\right )-\left ( \frac{3\sqrt{2}}{\sqrt{15}+3\sqrt{2} } \times\frac{\sqrt{15}-3\sqrt{2}}{\sqrt{15}-3\sqrt{2}}\right )
\Rightarrow \frac{7\sqrt{3}\left ( \sqrt{10} -\sqrt{3}\right )}{10-3}-\frac{2\sqrt{5}\left ( \sqrt{6}-\sqrt{5} \right )}{6-5}-\frac{3\sqrt{2}\left ( \sqrt{15}-3\sqrt{2} \right )}{15-8}
\left [ \because a^{2}-b^{2} = \left ( a+b \right )\left ( a-b \right )\right ]
\Rightarrow \frac{7\sqrt{3}\left ( \sqrt{10} -\sqrt{3}\right )}{7}-\frac{2\sqrt{5}\left ( \sqrt{6}-\sqrt{5} \right )}{1}-\frac{3\sqrt{2}\left ( \sqrt{15}-3\sqrt{2} \right )}{3}
\Rightarrow \frac{7\sqrt{30}-21}{7}-\frac{2\sqrt{30}-10}{1}+\frac{3\sqrt{30}-18}{3}
\Rightarrow \frac{21\sqrt{30}-63-42\sqrt{30}+210+21\sqrt{30}-126}{21}
\Rightarrow \frac{21}{21}= 1
Hence the answer is 1.

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