Evaluate the nested square root $x = \sqrt{11 - 2\sqrt{10}} - \sqrt{11 + 2\sqrt{10}}$

Question

Evaluate: $x = \sqrt{11 - 2\sqrt{10}} - \sqrt{11 + 2\sqrt{10}}$

You may have seen my other Q/A here, but I am finding a different way, with perhaps perfect squares.

If we seperate,

$y = \sqrt{11 - 2\sqrt{10}}$. But this doesn't help.

Answer 1

$$\begin{align}x&=\sqrt{11-2\sqrt{10}}-\sqrt{11+2\sqrt{10}}\\&=\sqrt{10+1-2\sqrt{10\times 1}}-\sqrt{10+1+2\sqrt{10\times 1}}\\&=\sqrt{(\sqrt{10}-\sqrt 1)^2}-\sqrt{(\sqrt{10}+\sqrt 1)^2}\\&=(\sqrt{10}-\sqrt 1)-(\sqrt{10}+\sqrt 1)\\&=-2\end{align}$$

Answer 2

HINT $x^2$ is easily computed using the difference of squares:

$$ (a+b)(a-b) = a^2 - b^2 $$

Answer 3

Another method using the formula: $$\sqrt { a+\sqrt { b } } =\sqrt { \frac { a+\sqrt { { a }^{ 2 }-b } }{ 2 } } +\sqrt { \frac { a-\sqrt { { a }^{ 2 }-b } }{ 2 } } \\ \sqrt { a-\sqrt { b } } =\sqrt { \frac { a+\sqrt { { a }^{ 2 }-b } }{ 2 } } -\sqrt { \frac { a-\sqrt { { a }^{ 2 }-b } }{ 2 } } $$ $$x=\sqrt { \frac { 11+\sqrt { 121-40 } }{ 2 } } -\sqrt { \frac { 11-\sqrt { 121-40 } }{ 2 } } -$$ $$\\ -\left[ \sqrt { \frac { 11+\sqrt { 121-40 } }{ 2 } } +\sqrt { \frac { 11-\sqrt { 121-40 } }{ 2 } } \right] =$$ $$=-2\sqrt { \frac { 11-\sqrt { 121-40 } }{ 2 } } =-2$$