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Can anyone explain why $a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(b*c)}$

Its been a while since I worked much with exponents, and I got confused and thought that $(a^b)^c = a^{(b^c)}$

I wasn't sure, so I tried my usual strategy of decompressing the equation (eg. $x^2 = (x)(x)...(x)$) but that didn't do the trick in this case (maybe I did it wrong). How can I show the difference between the two expressions in a simple and intuitive way?

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marked as duplicate by MJD, amWhy, Marvis, Henry, EuYu Nov 18 '12 at 22:37

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5 Answers

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Recall that if $y \in \mathbb{Z}^+$, then $$x^y = \underbrace{x \cdot x \cdot x \cdots x}_{y \text{ times}}$$ Hence, \begin{align} \left(a^b \right)^c & = \underbrace{a^b \cdot a^b \cdot a^b \cdots a^b}_{c \text{ times}}\\ & = \underbrace{\underbrace{a \cdot a \cdots a}_{b \text{ times}} \cdot \underbrace{a \cdot a \cdots a}_{b \text{ times}} \cdot \underbrace{a \cdot a \cdots a}_{b \text{ times}} \cdots \underbrace{a \cdot a \cdots a}_{b \text{ times}}}_{c \text{ times}}\\ & = \underbrace{a \cdot a \cdot a \cdots a}_{bc \text{ times}}\\ & = a^{bc} \end{align} Similarly, \begin{align} a^{(b^c)} & = \underbrace{a \cdot a \cdot a \cdots a}_{b^c \text{ times}} \end{align} Hence, $\left(a^b \right)^c$ is $a$ multiplied $bc$ times together, whereas $a^{(b^c)}$ is $a$ multiplied $b^c$ times together.

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For simplicity, assume that $b$ and $c$ are positive integers. Then $$(a^b)^c = \underbrace{a^b \cdot a^b \cdots a^b}_{c \text{ factors}} = \underbrace{\underbrace{a \cdot a \cdots a}_{b \text{ factors}} \cdot \underbrace{a \cdot a \cdots a}_{b \text{ factors}} \cdots \underbrace{a \cdot a \cdots a}_{b \text{ factors}}}_{c \text{ factors}}.$$

How many factors of $a$ do you have in the final expression?

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What is $(2^2)^3$?

What is $2^{(2^3)}$?

Frankly, there's no obvious reason why these two things should be equal. I'm not sure what you mean by "the usual strategy": if you have a specific rationale that makes you think those are the same, share it and we can figure out why it misleads you.

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For a quick check just take a simple base and exponent that you know and check it. For example, let $a=2$, $b=2$, and $c=3$. Now you have $(2^2)^3=64$ and $2^{2^3}=256$. So now you can clearly see that they aren't equal. Just be careful not to use something that does coincidently equal the same. For example I almost set $a,b,c$ all equal to 2 but realized they would all equal the same. So just try out a few different ones if you get them mixed up.

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$(a^b)^c = a^{bc}$. Now, you just have to convince yourself that $bc \neq b^c$ in general, and so $a^{bc} \neq a^{b^c}$ since the two expressions have different, unequal exponents.

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