# How do you simplify 16/(sqrt [5] - sqrt [7])?

Oct 16, 2015

$- 8 \left(\sqrt{5} + \sqrt{7}\right)$

#### Explanation:

To simplify this expression, you have to rationalize the denominator by using its conjugate expression.

For a binomial, you get its conjugate by changing the sign of the second term.

In your case, that would imply having

$\sqrt{5} - \sqrt{7} \to {\underbrace{\sqrt{5} \textcolor{red}{+} \sqrt{7}}}_{\textcolor{b l u e}{\text{conjugate}}}$

So, multiply the fraction by $1 = \frac{\sqrt{5} + \sqrt{7}}{\sqrt{5} + \sqrt{7}}$ to get

16/(sqrt(5) - sqrt(7)) * (sqrt(5) + sqrt(7))/(sqrt(5) + sqrt(7)) = (16(sqrt(5) + sqrt(7)))/((sqrt(5) - sqrt(7))(sqrt(5)+sqrt(7))

The denominator is now in the form

$\left(a - b\right) \left(a + b\right) = {a}^{2} - {b}^{2}$

This means that you can write

$\frac{16 \left(\sqrt{5} + \sqrt{7}\right)}{{\left(\sqrt{5}\right)}^{2} - {\left(\sqrt{7}\right)}^{2}} = \frac{16 \left(\sqrt{5} + \sqrt{7}\right)}{5 - 7} = \textcolor{g r e e n}{- 8 \left(\sqrt{5} + \sqrt{7}\right)}$