Question

Express $\frac{15}{\sqrt{10}+\sqrt{20}+\sqrt{40}-\sqrt{5}-\sqrt{80}}$ with a rational denominator:

• A. $\sqrt{10}+\sqrt{5}$
• B. $\sqrt{10}-\sqrt{5}$
• C. $-\sqrt{10}+\sqrt{5}$
• D. None of these

Answer 1

Answer: (A)

$\sqrt{10}+\sqrt{5}$

$\frac{15}{\sqrt{10}+\sqrt{20}+\sqrt{40}-\sqrt{5}-\sqrt{80}}$
$\Rightarrow \frac{15}{\sqrt{10}+\sqrt{4\times 5}+\sqrt{10\times 4}-\sqrt{5}-\sqrt{16\times 5}}$
$\Rightarrow \frac{15}{\sqrt{10}+2\sqrt{5}+2\sqrt{10}-\sqrt{5}-4\sqrt{5}}$
$\Rightarrow \frac{15}{\sqrt{10}+2\sqrt{10}+\sqrt{5}-4\sqrt{5}}$
$\Rightarrow \frac{15}{\sqrt{10}(1+2)+\sqrt{5}(1-4)}$
$\Rightarrow \frac{15}{3\sqrt{10}-3\sqrt{5}}\Rightarrow \frac{15}{3(\sqrt{10}-\sqrt{15})}$
$\Rightarrow \frac{5}{\sqrt{10}-\sqrt{5}}$
$\Rightarrow \frac{5}{\sqrt{10}-\sqrt{5}}\times \frac{\sqrt{10}+\sqrt{5}}{\sqrt{10}+\sqrt{5}}$
$\Rightarrow \frac{5(\sqrt{10}+\sqrt{5})}{10-5}\Rightarrow \sqrt{10}+\sqrt{5}$