When are the Laws of Exponents correct?

Question

The rules of powers are in highschool books often briefly stated in the following way:

1. $\displaystyle a^n \cdot a^m = a^{n+m}$
2. $\displaystyle \frac{a^n}{a^m} = a^{n-m}$
3. $\displaystyle \left (a\cdot b\right )^n = a^n \cdot b^n $
4. $\displaystyle \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
5. $\displaystyle \left(a^n\right )^m = a^{n\cdot m}$

I sometimes try to explain to my highschool students that those rules are not always true. For example, $0^{-2} \cdot 0^{2} = 0^0$ or I give other interesting false deductions such as: $$\left(-1\right)^3=(-1)^{6\cdot \frac{1}{2}}=\left((-1)^{6}\right)^{\frac{1}{2}}=\sqrt{1}=1 $$

However I could not find an exact reference to where those rules are true.

Steward's Review of Algebra states that those rules are true if $a$ and $b$ are positive (real) numbers, and $n$ and $m$ are rational numbers. This is of course very conservative. Those rules are also true if $a\ne$, $b\ne 0$ and $n,m$ integers. Besides that I think many of those rules are also true if $n,m$ are real numbers.

So my question is, when are the above rules correct?

Answer 1

Provided $a,b>0$, all the rules are true for real $a,b,m,n$.

If $a=0$ or $b=0$, no negative power may appear.

For $a<0$ or $b<0$, irrational exponents are excluded. Rational ones are possible provided the denominator of the simplified fraction is odd. This can cause rule 5 to fail ($(-1)^1\ne((-1)^{1/2})^2$).

Answer 2

Whenever the base is positive and the exponent is real, or the base is zero and the exponent is positive, or the base is negative and the exponent is an integer.