Question

What is the value of $\frac{1}{1+\sqrt{2}+\sqrt{3}}+\frac{1}{1-\sqrt{2}+\sqrt{3}}$?

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

Answer 1

The correct option is A $1$

The value of $\frac{1}{(1+\sqrt{3})+\sqrt{2}}+\frac{1}{(1+\sqrt{3})-\sqrt{2}}$ is
$=\frac{(1+\sqrt{3}-\sqrt{2})+(1+\sqrt{3}+\sqrt{2})}{(1+\sqrt{3}{)}^2-(\sqrt{2}{)}^2}$
$=\frac{2(1+\sqrt{3})}{1+3+2\sqrt{3}-2}$
$=\frac{2(1+\sqrt{3})}{2(1+\sqrt{3})}=1$