Question

Express $\frac{1}{\sqrt{10}+\sqrt{14}+\sqrt{15}+\sqrt{21}}$ with a rational denominator:

• A. $\frac{\sqrt{21}+\sqrt{10}+\sqrt{14}-\sqrt{15}}{2}$
• B. $\frac{\sqrt{21}+\sqrt{10}-\sqrt{14}+\sqrt{15}}{2}$
• C. $\frac{\sqrt{21}+\sqrt{10}-\sqrt{14}-\sqrt{15}}{2}$
• D. $\frac{\sqrt{21}-\sqrt{10}-\sqrt{14}-\sqrt{15}}{2}$

Answer 1

Answer: (C)

$\frac{\sqrt{21}+\sqrt{10}-\sqrt{14}-\sqrt{15}}{2}$

As given \frac{1}{\sqrt{10}+\sqrt{14}+\sqrt{15}+\sqrt{21}}
Multiply and divide by $\sqrt{10}+\sqrt{21}-\sqrt{14}-\sqrt{15}$
$\Rightarrow \frac{1}{\sqrt{10}+\sqrt{14}+\sqrt{15}+\sqrt{21}}\times \frac{\sqrt{10}+\sqrt{21}-\sqrt{14}-\sqrt{15}}{\sqrt{10}+\sqrt{21}-\sqrt{14}-\sqrt{15}}$
$\Rightarrow \frac{\sqrt{10}+\sqrt{21}-\sqrt{14}-\sqrt{15}}{10+21-14-15}$
$\Rightarrow \frac{\sqrt{10}+\sqrt{21}-\sqrt{14}-\sqrt{15}}{31-29}$
$\Rightarrow \frac{\sqrt{10}+\sqrt{21}-\sqrt{14}-\sqrt{15}}{2}$