# How do you simplify sqrt 8 (sqrt 13 - sqrt 117)?

Jul 29, 2016

$- 4 \sqrt{26}$

#### Explanation:

Consider the factor tree of 117

Note that if the sum of the digits is exactly divisible by 3 then the whole number is also divisible by 3.

So the prime number factors of 117 are ${3}^{2} \times 13$

Given:$\text{ } \sqrt{8} \left(\sqrt{13} - \sqrt{117}\right)$

Write as:

$\sqrt{8} \left(\sqrt{13} - 3 \sqrt{13}\right)$

Factor out $\sqrt{13}$ giving:

$\sqrt{8} \sqrt{13} \left(1 - 3\right)$

Note that $\sqrt{8} = 2 \sqrt{2}$ giving:

$2 \sqrt{2} \sqrt{13} \times \left(- 2\right)$

$\implies - 4 \sqrt{2 \times 13} \to - 4 \sqrt{26}$