# Question #46f3e

May 2, 2017

Let us assume a square root....

#### Explanation:

Assuming that we are simplifying a square root (rather than a cube root, fourth root, or other radical), we are looking for perfect squares that divide 108. It helps to factor 108 completely into primes.

$108 = 2 \cdot 54$

$108 = 2 \cdot 2 \cdot 27$

$108 = 2 \cdot 2 \cdot 3 \cdot 9$

FINALLY, we have:

$108 = 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3$

Since we are searching for perfect squares, any pair of identical factors IS a perfect square. Perfect squares extract as a single factor of their type.

$2 \cdot 2$ extracts as a $2$.

$3 \cdot 3$ extracts as a $3$.

The remaining $3$ stays under the radical.

So

$\sqrt{108} = 2 \cdot 3 \cdot \sqrt{3}$.

Multiply the factors that are outside.

$\sqrt{108} = 6 \sqrt{3}$