Question

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

Answer 1

`(40-5sqrt7)/57`

Explanation:

Since the equation `5/(8+sqrt7)` has a radical in the denominator, it must be rationalized. However, you cannot simply multiply the numerator and denominator by `sqrt7`, because the denominator would still remain as `8sqrt7 + 7`.

Instead, you must multiply by the conjugate of `8+sqrt7`, which is `8-sqrt7`. The conjugate of something is simply swapping the sign of the equation from either negative to positive, or vise versa.

Multiplying by the conjugate,

`5/(8+sqrt7) * (8-sqrt7)/(8-sqrt7)`

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:

`(40-5sqrt7)/57`

Which cannot be simplified any further.