Question

$\sqrt{4 - \sqrt{7 } } \div \sqrt{8 + 3 \sqrt{7} } + 2 \sqrt{2}$
solve this problem

Answer 1

$\frac{\sqrt{4 - \sqrt{7}}}{\sqrt{8+3\sqrt{7}}} + 2\sqrt{2}$

$\implies \frac{\sqrt{2(8-2\sqrt{7})}}{\sqrt{2(16 + 6\sqrt{7}}} + 2\sqrt{2}$

$\implies \frac{\sqrt{ 2 ( 7+ 1 - 2\sqrt{7})}}{\sqrt{2(3^2+7+6\sqrt{7})}}+2\sqrt{2}$

$\implies \frac{\sqrt{2(\sqrt{7 }- 1)^2}}{\sqrt{2(3+\sqrt{7})^2}}+2\sqrt{2}$

$\implies \frac{(\sqrt{7} - 1)\sqrt{2}}{(3+\sqrt{7})\sqrt{2}}+2\sqrt{2}$

$\implies \frac{\sqrt{7}-1 + 2\sqrt{2}(3+\sqrt{7})}{\sqrt{3 + \sqrt{7}}}$

$\implies \frac{\sqrt{7}-1 + 6\sqrt{2}+2\sqrt{14}}{\sqrt{3 + \sqrt{7}}}$

We need to factorise the numerator now

Then :

$\implies \frac{\sqrt{7}-1 + 6\sqrt{2}+2\sqrt{14}}{\sqrt{3 + \sqrt{7}}} \times \frac{3 - \sqrt{7}}{3 - \sqrt{7} }$

Sorry I cant figure out how to factorise the numerator!

If you factorise it then its done!

Now do it your self.

I just gave you the hint

Hope it helps