Question 406070
Simplifying Square-Root Terms
1.

To simplify a square root, you "take out" anything that is a "perfect square"; that is, you take out front anything that has two copies of the same factor:

{{{sqrt(16)=sqrt(4^2)=4}}}

When you solve the equation {{{x^2 = 4}}}, you are trying to find all possible values that might have been squared to get 4. But when you are just simplifying the expression{{{ sqrt(4)}}}, the ONLY answer is "{{{2}}}"; this {{{positive }}}result is called the "{{{principal}}}" {{{root}}}. 
Other roots, such as {{{-2}}}, can be defined using graduate-school topics like "complex analysis" and "branch functions".

2.
Sometimes the argument of a radical is {{{not}}} a {{{perfect}}}{{{ square}}}, but it may "contain" a square amongst its factors. 
To simplify, you need to factor the argument and "take out" anything that is a square; you find anything you've got a pair of inside the radical, and you {{{move}}}{{{ it}}}{{{ out }}}{{{front}}}. 

To do this, you use the fact that you can switch between the multiplication of roots and the root of a multiplication. In other words, radicals can be manipulated similarly to powers:



{{{(ab)^n=a^n*b^n }}}  

{{{root(n,ab)=root(n,a)*root(n,b)}}}