8.1 Exponentials

Hopefully you’re vaguely aware that $a^{b}$ means "a multiplied by itself b times" (for $b\in\mathbb{N}$)¹¹ 1 If you’re not sure about this notation, review the section on ”Sets and Numbers”. From this definition, there are a bunch of useful facts we can derive.

A somewhat non-rigorous argument for this being true is as follows: $a^{b}=a*a*a*...*a$ (a times itself b times). When we multiply $a^{b}$ by $a^{c}$, which is equal to $a^{c}=a*a*a*...*a$ (a times itself c times), we are then multiplying $a$ times itself $b$ times by $a$ times itself $c$ times. Overall, therefore we are multiplying $a$ by itself $b+c$ times.

To see that this is true, first note that we start by multiplying $a$ by itself $b$ times ($a*a*a*...*a$). We then raise this to the power of $c$, so $(a*a*a*...*a)^{c}$, which means we have $(a*a*a*...*a)*(a*a*a*...*a)*...*(a*a*a*...*a)$. In total, there are $c*b$ lots of $a$ (every bracket is $b$ lots of $a$, and there are $c$ of the brackets, so overall there are $c*b$ lots of $a$).