How do you simplify radical expressions?

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

Using the identities `\sqrt{a}^2=a` and `(a-b)(a+b)=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 `{3x}/{\sqrt{x+3}}`. If we multply this fraction by `{\sqrt{x+3}}/{\sqrt{x+3}}`, we won't change its value (since of course `{\sqrt{x+3}}/{\sqrt{x+3}}=1`, but we can rewrite it as follows:
`{3x}/{\sqrt{x+3}} \cdot {\sqrt{x+3}}/{\sqrt{x+3}}= \frac{3x\sqrt{x+3}}{\sqrt{x+3}^2}`, and finally obtain
`\frac{3x\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

`\frac{\cos(x)}{\sqrt{x}+\sqrt{\sin(x))} \cdot \frac{\sqrt{x}-\sqrt{\sin(x)}}{\sqrt{x}-\sqrt{\sin(x)}}` which is

`\frac{\cos(x)(\sqrt{x}-\sqrt{\sin(x)})}{\sqrt{x}^2-\sqrt{\sin(x)}^2}`

which finally equals

`\frac{\cos(x) (\sqrt{x}-\sqrt{\sin(x)})}{x-\sin(x)}`

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!