Behaviour of Exponents

Question

Something that many early students, including myself, take for granted is that $$x^\frac{3}{2}=\sqrt{x^3}=(\sqrt{x})^3$$

but is this true? Is exponentiation "commutative" and does a fractional exponent mean the same thing as a root?

Answer 1

No, this is not true in general.

You could create pointless results like this. Take for example $\sqrt{-2}$ which is not definied in $\mathbb{R}$.

But we have that $\sqrt{(-2)^2}=\sqrt{4}=2$, while $(\sqrt{-2})^2$ does not make sense.

Answer 2

In the reals it is true that $(a^b)^c=a^{bc}=(a^c)^b$. This works for all $a \gt 0$ and any real $b,c$, so fractions are included.

It is not true in the complex numbers because there are multiple roots of a number and you need to be careful to pick the correct one. We avoid the root sign in the complex field for that reason.