In this Khan Academy video $(a^3)^{^2}$ is interpreted as $a^6$. Considering the placement of the outer exponent, shouldn't it be $a^{3^2}=a^9$ instead? I thought $$ (a^m)^n=a^{mn} $$ but $$ (a^m)^{^n}=a^{m^n} $$
I have also seen $(a^m)^{^n}=a^{mn}$ in books.
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The font size should not be of such relevance - otherwise handwritten math would be totally ambiguous. Parentheses take precedence - always. Thus here we first compute $a^3$ and then square the result (which happens to be the same as $a^6$).
By the way, the fact that $(a^m)^n$ can be expressed simply as $a^{mn}$, whereas $a^{(m^n)}$ does not allow any similar rewriting, is the reason why we have the convention that nested exponentiation without parentheses is to be evaluated from right to left (or from top to bottom if you wish), i.e., $a^{m^n}:=a^{(m^n)}$.
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I think an easy was of looking at it would be by using its multiplicative form.for example $(a^2)^3$ actually means $(a^2)*(a^2)*(a^2) $which is nothing but $a$x$a$x$a$x$a$x$a$ which is a multiplication of 6 $a$s so it is $a^6$ whereas a^2^3 is actually a^ (2^3)so it is $a^8$ . Also as u mentioned in textbooks (a^2)^^3 written as $a^6$ it's probably just a printing error and they meant $(a^2)^3$
$(a^m)^n$ and $(a^m)^{^n}$ represent the same thing, and that is $a^{mn}$. The fact that in the second case the exponent $n$ is a bit higher does not change the operation nor the order in which they are applied.