# Multiplication and Division of Radicals

## Key Questions

• $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$

or $\sqrt{a} \sqrt{b} = \sqrt{a \times b}$

• When you have a root (square root for example) in the denominator of a fraction you can "remove" it multiplying and dividing the fraction for the same quantity. The idea is to avoid an irrational number in the denominator.
Consider:
$\frac{3}{\sqrt{2}}$
you can remove the square root multiplying and dividing by $\sqrt{2}$;
$\frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$
This operation does not change the value of your fraction because $\frac{\sqrt{2}}{\sqrt{2}} = 1$ anyway and your fraction does not change by multiplying $1$ to it.

Now you can multiply in the numerator and denominator:
$\frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{3 \cdot \sqrt{2}}{\left(\sqrt{2}\right) \cdot \left(\sqrt{2}\right)}$ giving:
$\frac{3 \cdot \sqrt{2}}{2}$ you have removed the square root from the denominator! (ok it went to the nominator but this is ok).

Now a problem for you; what happens when the root is not alone???!!!
If you have:
$\frac{3}{1 + \sqrt{2}}$???
You can use the same technique but...what do you use to multiply and divide?
HINT: look at what happens if you do this:
$\left(1 + \sqrt{2}\right) \cdot \left(1 - \sqrt{2}\right)$

• Multiplication

$\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$

Division

$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

I hope that this was helpful.