# How do you rationalize the denominator and simplify (5+ sqrt 5)/(8- sqrt 5)?

Apr 10, 2015

Multiply the numerator an denominator by the conjugate of the denominator, then look for simplification factors

$\frac{5 + \sqrt{5}}{8 - \sqrt{5}}$

$= \frac{5 + \sqrt{5}}{8 - \sqrt{5}} \cdot \frac{8 + \sqrt{5}}{8 + \sqrt{5}}$

$= \frac{40 + 13 \sqrt{5} + 5}{64 - 5}$

$= \frac{45 + 13 \sqrt{5}}{59}$

Apr 10, 2015

Multiply the fraction by $1$ in the form $\frac{8 + \sqrt{5}}{8 + \sqrt{5}}$

$\frac{\left(5 + \sqrt{5}\right)}{\left(8 - \sqrt{5}\right)} \frac{\left(8 + \sqrt{5}\right)}{\left(8 + \sqrt{5}\right)} = \frac{40 + 5 \sqrt{5} + 8 \sqrt{5} + 5}{64 - 5}$

$= \frac{45 + 13 \sqrt{5}}{59}$.

This works because of the product: $\left(a - b\right) \left(a + b\right) = {a}^{2} - {b}^{2}$.

If one or both ob $a$, $b$ have square roots, then the product doesn't: for example: $\left(a - \sqrt{c}\right) \left(a + \sqrt{c}\right) = {a}^{2} - {\left(\sqrt{c}\right)}^{2} = {a}^{2} - c$.