# How do you find the square root of -324/728?

Aug 14, 2016

$\sqrt{- \frac{324}{728}} = \frac{9 \sqrt{182}}{182} i \cong 0.66712 i$

#### Explanation:

$\sqrt{- \frac{324}{728}} = \sqrt{\frac{324}{728}} i$

All natural numbers can be expressed as the product of prime numbers. In evaluating the roots of rational numbers it is often useful to decompose the number into its prime factors.

Notice that: $324 = {2}^{2} \times {3}^{4}$
And that: $728 = {2}^{3} \times 7 \times 13$

Hence: $\sqrt{- \frac{324}{728}} = \frac{\sqrt{{2}^{2} \times {3}^{4}}}{\sqrt{{2}^{3} \times 7 \times 13}} i$

Since any pair of prime factors may be taken out of the sqrt

$\sqrt{- \frac{324}{728}} = \frac{2 \times 3 \times 3}{2 \times \sqrt{2 \times 7 \times 13}} i$

$= \frac{\cancel{2} \times 3 \times 3}{\cancel{2} \times \sqrt{2 \times 7 \times 13}} i$

$= \frac{9 \sqrt{2 \times 7 \times 13}}{2 \times 7 \times 13} i$

$= \frac{9 \sqrt{182}}{182} i \cong 0.66712 i$