Fractional Exponent Identity: Is the inside of a root necessarily calculated first?

Question

I know $a^{b/c}=\sqrt[c]{a^b}$, but does $a^b$ have to be calculated first, or does $a^{b/c}=(\sqrt[c]a\,)^b$ also work? Thanks.

Answer 1

I am assuming that $a > 0$ is a real number, and taking $$ a^{b/c} = \sqrt[c]{a^b} $$ as the definition of the rational power. Let's raise the new expression $x = (\sqrt[c]{a}\,)^b$ to the power $c$ and apply exponent properties: \begin{align} x^c &= \bigl((\sqrt[c]{a}\,)^b\bigr)^c \\ &= (\sqrt[c]{a}\,)^{bc} \\ &= (\sqrt[c]{a}\,)^{cb} \\ &= \bigl((\sqrt[c]{a}\,)^c\bigr)^b \\ &= a^b \end{align} Thus, $x$ is the $c$-root of $a^b$, which was your original definition for $a^{b/c}$, hence they are all the same number.

Note: This all becomes a bit more complicated if you allow $a$ to be a negative real number or non-real complex number or any other type of number where raising to a power is not a one-to-one operation. In that case, it's not even clear how to define the root since there's potentially more than one.