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

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

Title	square root of square root binomial
Canonical name	SquareRootOfSquareRootBinomial
Date of creation	2013-03-22 15:21:21
Last modified on	2013-03-22 15:21:21
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	10
Author	pahio (2872)
Entry type	Topic
Classification	msc 11A25
Related topic	TakingSquareRootAlgebraically
Related topic	SquareRootsOfRationals
Defines	square root binomial
Defines	square root polynomial