# How do you simplify radical expressions?

Feb 1, 2015

There are two common ways to simplify radical expressions, depending on the denominator.

Using the identities $\setminus {\sqrt{a}}^{2} = a$ and $\left(a - b\right) \left(a + b\right) = {a}^{2} - {b}^{2}$, in fact, you can get rid of the roots at the denominator.

Case 1: the denominator consists of a single root. For example, let's say that our fraction is $\frac{3 x}{\setminus \sqrt{x + 3}}$. If we multply this fraction by $\frac{\setminus \sqrt{x + 3}}{\setminus \sqrt{x + 3}}$, we won't change its value (since of course $\frac{\setminus \sqrt{x + 3}}{\setminus \sqrt{x + 3}} = 1$, but we can rewrite it as follows:
$\frac{3 x}{\setminus \sqrt{x + 3}} \setminus \cdot \frac{\setminus \sqrt{x + 3}}{\setminus \sqrt{x + 3}} = \setminus \frac{3 x \setminus \sqrt{x + 3}}{\setminus {\sqrt{x + 3}}^{2}}$, and finally obtain
$\setminus \frac{3 x \setminus \sqrt{x + 3}}{x + 3}$

Case 2: the denominator consists of a sum/difference of roots. If we multiply by the difference/sum of the roots, we'll have the same result as above. For example, if you have

\frac{\cos(x)}{\sqrt{x}+\sqrt{\sin(x)}

You'll multiply numerator and denominator by the difference \sqrt{x}-\sqrt{\sin(x), and obtain

$\setminus \frac{\setminus \cos \left(x\right)}{\setminus \sqrt{x} + \setminus \sqrt{\setminus \sin \left(x\right)}} \setminus \cdot \setminus \frac{\setminus \sqrt{x} - \setminus \sqrt{\setminus \sin \left(x\right)}}{\setminus \sqrt{x} - \setminus \sqrt{\setminus \sin \left(x\right)}}$ which is

$\setminus \frac{\setminus \cos \left(x\right) \left(\setminus \sqrt{x} - \setminus \sqrt{\setminus \sin \left(x\right)}\right)}{\setminus {\sqrt{x}}^{2} - \setminus {\sqrt{\setminus \sin \left(x\right)}}^{2}}$

which finally equals

$\setminus \frac{\setminus \cos \left(x\right) \left(\setminus \sqrt{x} - \setminus \sqrt{\setminus \sin \left(x\right)}\right)}{x - \setminus \sin \left(x\right)}$

Of course, when working with radicals, you always need to pay attention and make sure that the argument of the root is positive, otherwise you will write things that have no meaning!