Square root of both sides If you have the equation:
$x^2=2$
You get:
$x=\pm \sqrt{2}$
But what do you do actually do? What do you multiply both sides with to get this answer? You take the square root of both sides, but the square root of what? If you understand what i mean? 

The relevant theorem is

If $x$ denotes a real number and $y$ denotes a nonnegative real number, then the following are equivalent:
  
  
*
  
*$x^2 = y$
  
*$x = \sqrt{y}$ or $x = -\sqrt{y}$
  

Usually, people intuit this theorem through experience and/or mimicry, and are implicitly invoking it whenever they carry out an algebraic step like that.
It's not terribly hard to prove given enough other facts, as the other answers show.
