If $x = 5- \sqrt{21}$, find the value of $\dfrac {\sqrt x}{\sqrt{32-2x} - \sqrt{21}}$.

Question

PROBLEM: If $x = 5- \sqrt{21}$, find the value of $\dfrac {\sqrt x}{\sqrt{32-2x} - \sqrt{21}}$.

Solution:

$$x = 5- \sqrt{21}$$

$$\sqrt x = \sqrt {5- \sqrt{21}}$$

I am unable to continue from here.

Any assistance is appreciated.

Answer 1

You only need to notice that $$\sqrt{22+2\sqrt{21}}=\sqrt{21}+1;$$ $$\sqrt{10-2\sqrt{21}}=\sqrt{7}-\sqrt{3}.$$

In general, if $a\geqslant b\geqslant0$, $$\sqrt{a}\pm\sqrt{b}=\sqrt{a+b\pm2\sqrt{ab}}.$$ So, when you need to compute $\sqrt{x\pm\sqrt{y}}$, you could have a look if there is any simple solution for $x=a+b$, $y=4ab$.