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.

Quick beginner guide for asking a well-received question + please avoid "no clue" questions. What is your definition of $a^x?$ What were your attempts to prove your formula? I voted to close your post for lack of "context". — Anne Bauval, Jun 25 '23 at 15:19
However, here is an answer: https://math.stackexchange.com/a/634204/386889 — Anne Bauval, Jun 25 '23 at 15:22
This is one of the famous power laws and under the mentioned conditions , it is true. As mentioned , $a^{b^c}$ is interpreted different by convention. — Peter, Jun 25 '23 at 15:27

score 1 · Accepted Answer · answered Jun 25 '23 at 18:39

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

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

answered Jun 25 '23 at 18:39

Piita

965

Thank you! @Piita – Coco Jun 25 '23 at 21:21

The power of a power of a real number

1 Answers1