Roots of Negatives

Now every negative number has square roots. Split the number into a positive part and -1: since -9 is 9 times -1, $\sqrt{-9} = \sqrt{9}\cdot \sqrt{-1}$, which is 3 times i, or $3i$.

Check it by squaring: ${\left(3i\right)}^{2} = 9{i}^{2} = -9$. The rule in general: for a positive number a, $\sqrt{-a} = i\cdot \sqrt{a}$. So $\sqrt{-81} = 9i$ and $\sqrt{-144} = 12i$. The number comes first and the i after it.

Note that $\sqrt{-9}$ is not -3: ${\left(-3\right)}^{2}$ is 9, a positive number. The minus sign under the root is carried by the i.