# How do you rationalize the denominator and simplify [(x+3)^(1/2)-(x)^(1/2)] / 3?

Jan 27, 2018

$\frac{{\left(x + 3\right)}^{\frac{1}{2}} - {\left(x\right)}^{\frac{1}{2}}}{3} = \frac{1}{{\left(x + 3\right)}^{\frac{1}{2}} + {\left(x\right)}^{\frac{1}{2}}}$

#### Explanation:

The denominator is a natural number and it appears as you wanted to rationalize numerator. This is done as follows:

$\frac{{\left(x + 3\right)}^{\frac{1}{2}} - {\left(x\right)}^{\frac{1}{2}}}{3}$

= $\frac{{\left(x + 3\right)}^{\frac{1}{2}} - {\left(x\right)}^{\frac{1}{2}}}{3} \times \frac{{\left(x + 3\right)}^{\frac{1}{2}} + {\left(x\right)}^{\frac{1}{2}}}{{\left(x + 3\right)}^{\frac{1}{2}} + {\left(x\right)}^{\frac{1}{2}}}$

= $\frac{\left(x + 3\right) - \left(x\right)}{3 \left({\left(x + 3\right)}^{\frac{1}{2}} + {\left(x\right)}^{\frac{1}{2}}\right)}$

= $\frac{3}{3 \left({\left(x + 3\right)}^{\frac{1}{2}} + {\left(x\right)}^{\frac{1}{2}}\right)}$

= $\frac{1}{{\left(x + 3\right)}^{\frac{1}{2}} + {\left(x\right)}^{\frac{1}{2}}}$