how to simplify $ 2 \sqrt{2}\left(\sqrt{9-\sqrt{77}} \right) $

Question

How to simplify

$$ 2 \sqrt{2}\left(\sqrt{9-\sqrt{77}} \right) $$

so that it has no nested radicals? This question is same as that already posted but with a different point of view.

Answer 1

$$ 2\sqrt{2}\sqrt{9-\sqrt{77}}=2\sqrt{2}\left( \sqrt{\frac{9+2}{2}}-\sqrt{\frac{9-2}{2}}\right)=2\sqrt{11}-2\sqrt{7} $$

By the formula: $$ \sqrt{a\pm \sqrt{b}}=\sqrt{\dfrac{a+ \sqrt{a^2-b}}{2}}\pm\sqrt{\dfrac{a- \sqrt{a^2-b}}{2}} $$ that can easily verified and works well when $a^2-b$ is a perfect square.

You can see my answer to Denesting a square root: $\sqrt{7 + \sqrt{14}}$

Answer 2

$$2^{3/2} \bigl(9-77^{1/2}\bigr)^{1/2}= 2\bigl(18-2\cdot77^{1/2}\bigr)^{1/2}=2\Bigl(\bigl(11^{1/2}-7^{1/2}\bigr)^2\Bigr)^{1/2}=2(11^{1/2}-7^{1/2}).$$

Answer 3

$$2\sqrt{2}\sqrt{9-\sqrt{77}} = 2\sqrt{18-2\sqrt{7\cdot 11}} = 2\sqrt{(\sqrt{11}-\sqrt{7})^2} = 2\sqrt{11}-2\sqrt{7}.$$