Question

Rationalize and simplify the expression
$\frac{1}{2+\sqrt{3}}+\frac{1}{2-\sqrt{5}}+\frac{2}{\sqrt{5}-\sqrt{3}}$

• A. $\sqrt{2}+\sqrt{3}+\sqrt{5}$
• B. $1+\sqrt{5}$
• C. $1+\sqrt{3}$

Answer 1

$\frac{1}{2+\sqrt{3}}+\frac{1}{2-\sqrt{5}}+\frac{2}{\sqrt{5}-\sqrt{3}}$
$=\frac{1}{2+\sqrt{3}}\times \frac{2-\sqrt{3}}{2-\sqrt{3}}+\frac{1}{2-\sqrt{5}}\times \frac{2+\sqrt{5}}{2+\sqrt{5}}+\frac{2}{\sqrt{5}-\sqrt{3}}\times \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}Now,(a-b)\times (a+b)=a^2-b^2\Rightarrow \frac{2-\sqrt{3}}{4-3}+\frac{2+\sqrt{5}}{4-5}+2\times \left(\frac{\sqrt{5}+\sqrt{3}}{5-3}\right)\Rightarrow \frac{2-\sqrt{3}}{1}+\frac{2+\sqrt{5}}{-1}+2\times \left(\frac{\sqrt{5}+\sqrt{3}}{2}\right)\Rightarrow 2-\sqrt{3}+(-2-\sqrt{5})+\sqrt{5}+\sqrt{3}=0$