square root of square root binomial

Some people call the expressions of the form  $a+b\sqrt{c}$  the , especially when $c$ is an square-free integer greater than 1 (and $a$ and $b$ rational numbers).  On the high school (see e.g. division), or also polynomials containing several square root .  Taking the square root of a square root binomial is more difficult and usually results nested square roots.  However, there are some exceptions if the numbers are appropriate.  We have the

$$\sqrt{a+\sqrt{b}}=\sqrt{\frac{a+\sqrt{a^2-b}}{2}}+\sqrt{\frac{a-\sqrt{a^2-b}}{2}}$$

and

$$\sqrt{a-\sqrt{b}}=\sqrt{\frac{a+\sqrt{a^2-b}}{2}}-\sqrt{\frac{a-\sqrt{a^2-b}}{2}}.$$

If  $a^2-b$  happens to be square of a rational number, then the into expressions without nested square roots.

For example, because  $6^2-20=16=4^2$,  we obtain

$$\sqrt{6+2\sqrt{5}}=\sqrt{6+\sqrt{20}}=\sqrt{\frac{6+4}{2}}+\sqrt{\frac{6-4}{2}}=1+\sqrt{5},$$

and because  $4^2-7=9=3^2$,  we get

$$\sqrt{4-\sqrt{7}}=\sqrt{\frac{4+3}{2}}-\sqrt{\frac{4-3}{2}}=\frac{\sqrt{7}-1}{\sqrt{2}}=\frac{\sqrt{14}-\sqrt{2}}{2}.$$