# How do you rationalize the denominator (5 sqrt 6)/(sqrt 10)?

Apr 3, 2015

If you multiply both the numerator and the denominator by $\sqrt{10}$
then the denominator will become a non-radical.

$\frac{5 \sqrt{6}}{\sqrt{10}}$

$= \frac{5 \sqrt{6}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

$= \frac{5 \sqrt{60}}{10}$

which can be simplified as
$\frac{5 \cdot \left(2\right) \sqrt{15}}{10}$

$= \sqrt{15}$

Apr 3, 2015

Two methods:

Method 1

Multiply by $1$ in the form $\frac{\sqrt{10}}{\sqrt{10}}$ to get:

$\frac{5 \sqrt{6}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{5 \sqrt{60}}{10}$

Now simplify $\sqrt{60} = \sqrt{4 \cdot 15} = 2 \sqrt{15}$ So we have

$\frac{5 \sqrt{6}}{\sqrt{10}} = \frac{5 \sqrt{60}}{10} = \frac{5 \cdot 2 \sqrt{15}}{10} = \sqrt{15}$

Method 2

Notice that 6 and 10 are both divisible by 2, so we can start by simplifying:

$\frac{5 \sqrt{6}}{\sqrt{10}} = \frac{5 \sqrt{3} \sqrt{2}}{\sqrt{5} \sqrt{2}} = \frac{5 \sqrt{3}}{\sqrt{5}}$

Method 2a: now multiply by $1$ in the form $\frac{\sqrt{5}}{\sqrt{5}}$ to get:

$\frac{5 \sqrt{6}}{\sqrt{10}} = \frac{5 \sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{5 \sqrt{15}}{5} = \sqrt{15}$

Method 2b Reduce $\frac{a}{\sqrt{a}} = \sqrt{a}$ , So $\frac{5}{\sqrt{5}} = \sqrt{5}$
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$\frac{5 \sqrt{6}}{\sqrt{10}} = \frac{5 \sqrt{3}}{\sqrt{5}} = \frac{{\left(\sqrt{5}\right)}^{2} \sqrt{3}}{\sqrt{5}} = \sqrt{5} \sqrt{3} = \sqrt{15}$