Question

$\frac{4}{1-\sqrt{2}-\sqrt{3}}=$ ____________.

• A. $2-\sqrt{2}-\sqrt{6}$
• B. $2-\sqrt{2}-\sqrt{3}$
• C. 4
• D. $2-\sqrt{2}+\sqrt{6}$

Answer 1

The correct option is A

$2-\sqrt{2}-\sqrt{6}$

$\frac{4}{1-\sqrt{2}-\sqrt{3}}\times \frac{\left[\right(1-\sqrt{2})+\sqrt{3}]}{\left[\right(1-\sqrt{2})+\sqrt{3}]}$

$=\frac{4[1-\sqrt{2}+\sqrt{3}]}{(1-\sqrt{2}{)}^2-(\sqrt{3}{)}^2}$

$=\frac{4(1-\sqrt{2}+\sqrt{3})}{1-2\sqrt{2}+2-3}$

$=\frac{4(1-\sqrt{2}+\sqrt{3})}{-2\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}$

$=\frac{\text{/}{4}^1(1-\sqrt{2}+\sqrt{3})\left(\sqrt{2}\right)}{-\text{/}{4}_1}$

$=-\sqrt{2}+2-\sqrt{6}$

$=2-\sqrt{2}-\sqrt{6}$