radicals - What rational numbers have rational square roots?

All rational numbers have the fraction form

$$\frac a b,$$ where a and b are integers($b\neq0$).

My question is: for what $a$ and $b$ does the fraction have rational square root? The simple answer would be when both are perfect squares, but if two perfect squares are multiplied by a common integer $n$, the result may not be two perfect squares. Like:

$$\frac49 \to \frac 8 {18}$$

And intuitively, without factoring, $a=8$ and $b=18$ must qualify by some standard to have a rational square root.

Once this is solved, can this be extended to any degree of roots? Like for what $a$ and $b$ does the fraction have rational $n$th root?