## Breaking Down Square Roots of Large Numbers

Break the number into its factors and multiply the square roots of those.
$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$

## Fractions

General rule: $\sqrt{\frac a b} = \frac{\sqrt{a}}{\sqrt{b}}$

e.g. $\sqrt{\frac{9}{10}} = \frac{\sqrt{9}}{\sqrt{10}} = \frac{3}{\sqrt{10}}$

However solutions never contain square roots in a denominator. Therefore the denominator has to be rationalized. Since $\frac{\sqrt{10}}{\sqrt{10}} = 1$, this process is as simple as multiplying the result with a fraction that contains its denominator on both sides:

$\frac{3}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{3\sqrt{10}}{10}$

This works in general because $\sqrt{x} \times \sqrt{x} = x$

Also fractions should be simplified where possible e.g. $\sqrt{\frac{40}{2}} = \sqrt{20}$

### Square Root Term in Polynomial Denominator

Given a fraction such as $\frac {2}{\sqrt{2}-1}$, forming a difference of squares gets rid of the root in the denominator:

$a^2-b^2=(a+b)(a-b)$

So

$\frac {2(\sqrt{2}+1)}{2-1}$

## Misc Notes

$\sqrt{x^2} = x$