The following rules apply for exponents that are positive integers. The justification is sketched here.

These rules are also said to apply for exponents that are not positive integers–that is, zero and negative integers. If $p$ and $q$ are rational numbers, they are also said to apply. Finally, if $p$ and $q$ are irrational numbers, they are said to hold if $x$ and $y$ are positive–the other cases are not important for my purposes [1].

References

[1] Konrad Knopp. Theory and Application of Infinite Series. Dover Publications. 1990.