# How do you simplify 5/(8+square root of 7)?

Mar 28, 2018

$\frac{40 - 5 \sqrt{7}}{57}$

#### Explanation:

Since the equation $\frac{5}{8 + \sqrt{7}}$ has a radical in the denominator, it must be rationalized. However, you cannot simply multiply the numerator and denominator by $\sqrt{7}$, because the denominator would still remain as $8 \sqrt{7} + 7$.

Instead, you must multiply by the conjugate of $8 + \sqrt{7}$, which is $8 - \sqrt{7}$. The conjugate of something is simply swapping the sign of the equation from either negative to positive, or vise versa.

Multiplying by the conjugate,

$\frac{5}{8 + \sqrt{7}} \cdot \frac{8 - \sqrt{7}}{8 - \sqrt{7}}$

Since the denominator consists of two binomials, you must FOIL it. The 5 on the numerator can simply be distributed. If you are unsure as to what FOILing is, reference this: https://en.wikipedia.org/wiki/FOIL_method

After the previous step, you are left with:

$\frac{40 - 5 \sqrt{7}}{57}$

Which cannot be simplified any further.