# How do you rationalize the denominator and simplify (5sqrt6)/sqrt10?

##### 3 Answers
Mar 9, 2018

$\sqrt{15}$

#### Explanation:

multiply by $\sqrt{10}$:

$\frac{5 \sqrt{6} \sqrt{10}}{{\left(\sqrt{10}\right)}^{2}}$

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

${\left(\sqrt{10}\right)}^{2} = 10$

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

$\sqrt{60} = \sqrt{4} \cdot \sqrt{15} = 2 \sqrt{15}$

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

$\frac{5 \sqrt{60}}{10} = \frac{10 \sqrt{15}}{10}$

$= \frac{\sqrt{15}}{1}$

$= \sqrt{15}$

Mar 9, 2018

$\sqrt{15}$

#### Explanation:

In order to rationalize the denominator, you can multiply by $\frac{\sqrt{10}}{\sqrt{10}}$. This is the same as multiplying the fraction by one. If you multiply $\frac{5 \sqrt{6}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$, you get $\frac{5 \sqrt{60}}{10}$. If you multiply the square roots in the denominator, you get $\sqrt{100}$, which is equivalent to 10.

With $\frac{5 \sqrt{60}}{10}$, you can simplify to just $\frac{\sqrt{60}}{2}$.

Next, you can simplify $\sqrt{60}$ by doing a factor tree. When you do a factor tree, you will find you can pull out a factor of 2 from the square root leaving you with $\frac{2 \sqrt{15}}{2}$.

Lastly, just cancel out the 2 in the numerator and denominator, and you get the answer of $\sqrt{15}$

Mar 9, 2018

$\sqrt{15}$

#### Explanation:

$\text{using the "color(blue)"laws of radicals}$

•color(white)(x)sqrtaxxsqrtbhArrsqrtab

•color(white)(x)sqrtaxxsqrta=a

$\text{To rationalise the denominator that is eliminate the }$
$\text{radical from the denominator}$

$\text{multiply numerator/denominator by } \sqrt{10}$

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

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

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

$= \frac{5 \times \sqrt{4 \times 15}}{10}$

$= \frac{5 \times \sqrt{4} \times \sqrt{15}}{10}$

$= \frac{5 \times 2 \times \sqrt{15}}{10} = \frac{\cancel{10} \sqrt{15}}{\cancel{10}} = \sqrt{15}$