# How do you simplify 5/(4 + sqrt5)?

Jul 18, 2017

See a solution process below:

#### Explanation:

To simplify this express we need to rationalize the denominator. Or, in other words, remove the radicals from the denominator. We can do this by multiplying by the appropriate form of $1$:

$\frac{4 - \sqrt{5}}{4 - \sqrt{5}} \times \frac{5}{4 + \sqrt{5}} \implies \frac{5 \left(4 - \sqrt{5}\right)}{{4}^{2} + 4 \sqrt{5} - 4 \sqrt{5} + {\left(\sqrt{5}\right)}^{2}} \implies$

$\frac{\left(5 \times 4\right) - \left(5 \times \sqrt{5}\right)}{16 + \left(4 \sqrt{5} - 4 \sqrt{5}\right) + 5} \implies$

$\frac{20 - 5 \sqrt{5}}{16 + 0 + 5} \implies$

$\frac{20 - 5 \sqrt{5}}{21}$

Or

$\frac{20}{21} - \frac{5 \sqrt{5}}{21}$