Do all the properties of exponents work for every real exponent?

Question

I am writing an essay, but i doubt at typing that the properties of exponents works for every Real, I know it works for every Rational number.

$$\forall a,n,m\in\mathbb R$$ or $$\forall a\in\mathbb R, n,m\in \mathbb Q$$ ¿Which one is the correct For the next properties? :

$a^n a^m=a^{n+m}$

$\frac{a^n}{a^m} =a^{n-m}$

$(a^n)^m=a^{n \cdot m}$

$(a \cdot b)^n=a^n \cdot b^m$

$(\frac{a}{b})^n=\frac{a^n}{b^n} $

$a^0=1$

$a^1=a$

$a^{-n}=\frac{1}{a^n}$

Answer 1

Assuming $a$ and $b$ are positive all of these properties are true for any real numbers $n$ and $m$.

When $b = 0$ the properties involving division by zero are obviously meaningless; similarly, in the last property, raising 0 to a negative power is meaningless. Also, $0^0$ is subtle; it is best to define that as $1$ rather than $0$ (see "Concrete Mathematics").

When $a$ is negative, taking a negative number to an irrational power has problems, so although the properties still formally hold, you would need to be very carefull in using them.

Answer 2

It's best to restrict the bases to the positive reals. To see what can go wrong with negative bases, consider what happens when we use fractional exponents. Indeed, taking $a = -1$ and $m = 1/2$ and $n = 2$ yields: $$ (a^n)^m = ((-1)^2)^{1/2} = (1)^{1/2} = 1 \neq -1 = (-1)^1 = (-1)^{2 ~\cdot~ 1/2} = a^{n \cdot m} $$