What would $2^{3^4}$ equate to? I can think of two rules that may apply:
$a^{b^c} = a^{(b^c)}$ (Making $2^{3^4} = 2^{81}\approx2.417\cdot10^{24}$)
or
$a^{b^c} = (a^b)^c = a^{bc}$ (Making $2^{3^4} = 2^{12} = 4096$)
Which of these is true?
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2The first one is true. The power rule is $a ^{b^c}=a^{(b^c)}$. See here, section "Identities and properties". – Dietrich Burde Jun 19 '15 at 11:33
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As Dietrich Burde stated in a comment, the standard convention is that $a^{b^c} = a^{\left(b^c\right)}$. In my mind this convention is due to the fact that $\left(a^b\right)^c$ doesn't need the nested exponent since it is equal to $a^{bc}$ as you observe.
Ben Blum-Smith
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