For positive integers, $a \times b=\sum\limits^{b}{a}$, correct?
So therefore exponentiation where n is also a positive integer should be something like $a^n=\sum\limits^n{\sum\limits^a{a}}$
This is for a proof by induction, and I just want to see if I can simplify the proof enormously by doing this.
Is this correct, or am I missing something?
$\displaystyle\sum^a a = a^2$
$\displaystyle\sum^n a^2 = na^2$
You need $\displaystyle \sum^a \sum^a \sum^a \sum^a ... \sum^a$ n times to get $a^n$.
$$\displaystyle \sum^a 1= a^1$$
$$\displaystyle \sum^a\sum^a 1= a^2$$ $$\displaystyle \sum^a\sum^a\sum^a 1= a^3 \\ \vdots$$
Consider the case $a = 2$ if you would like a concrete example. Consider the case $a = 1$ if you want a counter example to your formula.