Square root of $8^3$ I'm only in intermediate algebra. I know that $\sqrt{8^3}$ is equal to $16\sqrt{2}$ but could you simply explain the process on how to get to that?
We can reduce $8^3$ to its prime factors:
$$8^3=512\implies 2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2$$
Because this is a square root, we are looking for groups of $2$. Similarly, if this was a cube root we would look for groups of 3:
 $$\boxed{2\cdot2}\boxed{2\cdot2}\boxed{2\cdot2}\boxed{2\cdot2}2$$
We have 4 groups of 2 which we will take out of the radical:
$$2\cdot2\cdot2\cdot2\sqrt{2}$$
We can now simplify this as:
$$\boxed{16\sqrt{2}}$$
I think a key insight is the exponent rule:

$$\sqrt {8^3}=$$ 

$$\sqrt {(2^3)^3}=$$

$$\sqrt {2^9}=$$

$${2^4} \cdot \sqrt{2}$$

$$=16\sqrt{2}$$