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How do you calculate the exponent of an exponent? In what order do you calculate the exponents?

For example, to calculate

${2^3}^4$

Is it

$({2^3})^4 = 8^4$

or

$2^{3^4} = 2^{81}$


ADDED: Say I'm given $y=x^2$ and then told that $x = m^3$. Can I say that in this case $y = m^9$?

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  • $\begingroup$ The latter. ${}{}{}$ $\endgroup$ – abiessu Dec 21 '15 at 21:22
  • $\begingroup$ $$2^{3^4}=2^{(3^4)}$$ $\endgroup$ – Dr. Sonnhard Graubner Dec 21 '15 at 21:23
  • $\begingroup$ You really should not write $a^{b^c}$ because $a^{(b^c)}$ does not usually equal $(a^b)^c$. So the notation is ambiguous. $\endgroup$ – Gregory Grant Dec 21 '15 at 21:27
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    $\begingroup$ No, $(m^3)^2=m^6$ not $m^9$. $\endgroup$ – Gregory Grant Dec 21 '15 at 21:28
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We usually define the notation $$ x^{y^z} $$ to mean $$ x^{\left(y^z\right)} $$

Mostly, this is because because this definition is most useful. Note that because of a power rule, if we wanted to write: $$ {\left(x^y\right)}^z$$

Then it's easier to write the equivalent $$ x^{yz}$$

Note that if $x=m^3$ and $y=x^2$, we can do the subsitution $y={\left(m^3\right)}^2=m^6$. It's important not to forget the brackets around $m^3$. This is similar to substituting, say, $b=a+1$ into $c=2b$. We must make sure to write $c=2(a+1)$ and not $c=2a+1$, which would be incorrect.


As a footnote, notation in mathematics is not always black and white. The most useful definition in one context could be a terrible definition in another context. Some may prefer that non-associative operations like exponentiation be parenthesized always, making $a^{b^c}$ incorrect notation. In my experience, however, most people will understand $a^{b^c}=a^{\left(b^c\right)}$ correctly without clarification.

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  • $\begingroup$ Chapter 11 of Niven's The Mathematics of Choice has a nice discussion of this non-associative product. $\endgroup$ – Trurl Dec 21 '15 at 22:12

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