Factor $-200=\left(10i\right)^{2}\times 2$. Rewrite the square root of the product $\sqrt{\left(10i\right)^{2}\times 2}$ as the product of square roots $\sqrt{\left(10i\right)^{2}}\sqrt{2}$. Take the square root of $\left(10i\right)^{2}$.
$$-\frac{1}{25}\times \left(10i\right)\sqrt{2}$$
Multiply $-\frac{1}{25}$ and $10i$ to get $-\frac{2}{5}i$.
