# How do you simplify fractions with square roots?

See below:

#### Explanation:

Let's say I have:

$\frac{5}{\sqrt{2}}$

How do I simplify this?

Remember that $\sqrt{2} \times \sqrt{2} = 2$ And so if I can multiply the denominator by $\sqrt{2}$, we'll get 2 for the denominator. And that can be done, so long as we multiply both the numerator and denominator by the same thing (which means we're multiplying the fraction by 1 and not changing its value):

$\frac{5}{\sqrt{2}} \left(1\right) = \frac{5}{\sqrt{2}} \left(\frac{\sqrt{2}}{\sqrt{2}}\right) = \frac{5 \sqrt{2}}{2}$

I can do the same type of thing if I have a mixed radical in my denominator. Let's say it's:

$\frac{5}{3 - \sqrt{2}}$

If I multiply two types of this kind of number, one adding the square root and the other subtracting, I can simplify. Let's look at our example again. With $3 - \sqrt{2}$, we can multiply by $3 + \sqrt{2}$:

$\left(3 - \sqrt{2}\right) \left(3 + \sqrt{2}\right) = 9 - 3 \sqrt{2} + 3 \sqrt{2} - 2 = 7$

And so for the full fraction, we have:

$\frac{5}{3 - \sqrt{2}} \left(\frac{3 + \sqrt{2}}{3 + \sqrt{2}}\right) = \frac{15 + 5 \sqrt{2}}{7}$