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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$\begingroup$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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