Under what conditions can the exponent product rule $a^{nm}=(a^n)^m$ be used.

Question

Under what conditions can you use the exponent product rule $a^{nm}=(a^n)^{^m}$? For example $i=\sqrt{-1}=(-1)^{1/2}=(-1)^{2\times 1/4}=((-1)^2)^{1/4}=(1)^{1/4}=1$ What's going wrong here?

Answer 1

The rule can be used when $a$ is real and positive. As you have noted, negative $a$ gives problems, and complex $a$ are no easier. Also, $m,n$ should be real numbers.

If $m,n$ are integers, then the rule holds for all complex, non-zero $a$. If $m,n$ are non-negative (but not necessarily integers), then the rule is valid for $a=0$ (I like setting $0^0=1$, but $0^0=0$ also works in this case).

Answer 2

What's wrong in your string of equalities is that:

• $i=\sqrt{-1}$ is wrong since $i$ is one of the square roots of $-1$ (the other one being $-i$, of course);
• $(-1)^{2\times(1/4)}=\bigl((-1)^2\bigr)^{1/4}$ is wrong because $(-1)^{2\times(1/4)}$ can be $i$ or $-i$, whereas $\bigl((-1)^2\bigr)^{1/4}$ can be one of four different numbers ($\pm1$ and $\pm i$);
• $1^{1/4}=1$ is wrong because $1^{1/4}$ can be any fourth root of $1$: again, $\pm1$ and $\pm i$.