# How do you simplify this? " "sqrt(3) + sqrt(1/3)

May 21, 2018

$\sqrt{3} + \sqrt{\frac{1}{3}} = \frac{4}{\sqrt{3}}$

#### Explanation:

$\sqrt{3} + \sqrt{\frac{1}{3}}$

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

$\sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}}$

$\sqrt{3} + \sqrt{\frac{1}{3}} = \frac{3}{\sqrt{3}} + \frac{1}{\sqrt{3}}$
$= \frac{3 + 1}{\sqrt{3}} = \frac{4}{\sqrt{3}}$
Thus,

$\sqrt{3} + \sqrt{\frac{1}{3}} = \frac{4}{\sqrt{3}}$

May 21, 2018

$= \frac{4}{3} \sqrt{3}$

#### Explanation:

You can write the square root of the fraction as separate roots:

$\sqrt{3} + \textcolor{b l u e}{\sqrt{\frac{1}{3}}} = \sqrt{3} + \textcolor{b l u e}{\frac{\sqrt{1}}{\sqrt{3}}}$

$\sqrt{3} + \frac{1}{\sqrt{3}} \text{ } \leftarrow$ rationalise the denominator

$= \sqrt{3} + \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$

$= \sqrt{3} + \frac{\sqrt{3}}{3} \text{ } \leftarrow$ factor out $\sqrt{3}$

$= \sqrt{3} \left(1 + \frac{1}{3}\right)$

$= \frac{4}{3} \sqrt{3}$

May 21, 2018

$\sqrt{3} + \sqrt{\frac{1}{3}}$ is not an equation and, therefore, cannot be solved but it can be rationalized.

#### Explanation:

$\sqrt{3} + \sqrt{\frac{1}{3}}$

$\sqrt{3} + \frac{\sqrt{1}}{\sqrt{3}}$

Multiply the second term by 1 in the form of $\frac{\sqrt{3}}{\sqrt{3}}$

$\sqrt{3} + \frac{\sqrt{1}}{\sqrt{3}} \frac{\sqrt{3}}{\sqrt{3}}$

Because of $\sqrt{3} \sqrt{3} = 3$: the denominator becomes 3:

$\sqrt{3} + \frac{\sqrt{3}}{3}$

Multiply the first term by 1 in the form of $\frac{3}{3}$:

$\frac{3 \sqrt{3}}{3} + \frac{\sqrt{3}}{3}$

The two fractions can be combined over the common denominator:

$\frac{3 \sqrt{3} + \sqrt{3}}{3}$

Add the terms in the numerator

$\frac{4 \sqrt{3}}{3}$

The above is the rationalized and simplified form.