How do you simplify 9/(6-sqrt8)?

Jul 19, 2017

Multiply the top and bottom by $\left(6 + \sqrt{8}\right)$ to make a difference of two squares and remove the square root.

Explanation:

The difference of two squares:
$\left(a + b\right) \left(a - b\right) = {a}^{2} - a b + a b - {b}^{2} = {a}^{2} - {b}^{2}$

$= \frac{9 \left(6 + \sqrt{8}\right)}{\left(6 - \sqrt{8}\right) \left(6 + \sqrt{8}\right)}$

$= \frac{54 + 9 \sqrt{8}}{{6}^{2} - {\left(\sqrt{8}\right)}^{2}}$

$= \frac{54 + 9 \sqrt{8}}{36 - 8}$

$= \frac{54 + 9 \sqrt{8}}{28}$

This is now simplified, as a fraction can have a root on top, but not on the bottom (as then it is irrational).

You could also round this value to 2.84, but the answer calculated is exact.