The power of a power of a real number

Question

I wonder that if $a$ is a positive real number and $b,c$ are real numbers, is it true that $$(a^b)^c = a^{bc}?$$

I only need this fact to prove a nonconstructive problem, so I didn't spend much time learning the real power of a real power. An answer is appreciated.

Answer 1

The following identities, often called exponent rules, hold for all positive real numbers, provided that the base is non-zero:

$a^{b+c}=a^{b}\times a^{c}\tag1$
$(a^{b})^{c}=a^{b\times c}\tag2$
$(a\times b)^{c}=a^{c}\times b^{c}\tag3$

To prove $(2)$, follow

$$(a^b)^c=e^{c\times\ln(a^b)}=e^{b\times c\times \ln(a)}=e^{\ln(a)\times (b\times c)}={a}^{b\times c}.$$