How to show that some nested radicals simplify to other

Question

Show that $\sqrt{8}\cdot\sqrt{9-\sqrt{77}}=2\cdot\sqrt{11}-2\cdot\sqrt{7}$

I have tried multiplying the radicals but that didn't work.

The resulting radicals do not add up or get subtracted.

I have tried taking commons also but that also doesn't work.

Answer 1

$\textbf {HINT:}$ Try to show that $$\big( \sqrt 8 \cdot \sqrt{9-\sqrt{77}} \big)^2 =\big(2\sqrt{11}-2\sqrt7\big)^2$$

Answer 2

multiply both side of the fraction to $\sqrt {2}$ $$\sqrt { 8 } \cdot \sqrt { 9-\sqrt { 77 } } =\sqrt { 8 } \frac { \sqrt { 18-2\sqrt { 77 } } }{ \sqrt { 2 } } =\sqrt { 8 } \frac { \sqrt { { \left( \sqrt { 7 } -\sqrt { 11 } \right) }^{ 2 } } }{ \sqrt { 2 } } =\frac { \sqrt { 8 } }{ \sqrt { 2 } } \left| \sqrt { 7 } -\sqrt { 11 } \right| =2\left( \sqrt { 11 } -\sqrt { 7 } \right) $$