Not a Perfect Square

When the number under the root is not a perfect square, do two things: take out the i, then simplify the root as in Algebra. For $\sqrt{-8}$, -8 is 8 times -1, so the root is $\sqrt{8}$ times i. And 8 is 4 times 2 with 4 a perfect square, so $\sqrt{8} = 2\sqrt{2}$.

That gives $\sqrt{-8} = 2\sqrt{2}i$, with the i written after the closed root. Writing $2\sqrt{2i}$ would put the i under the root, which is a different number. Check by squaring: ${\left(2\sqrt{2}i\right)}^{2}$ is 4 times 2 times ${i}^{2}$, so -8.

Terms with the same root and an i are like terms, and collect as 2x and x do: $\sqrt{-8}+\sqrt{-2}$ is $2\sqrt{2}i+\sqrt{2}i$, which is $3\sqrt{2}i$. When no perfect square divides the number, the root stays as it is: $\sqrt{-7} = \sqrt{7}i$.