8.1 Exponentials

Hopefully you’re vaguely aware that aba^{b} means "a multiplied by itself b times" (for bb\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.

ab+c=abaca^{b+c}=a^{b}a^{c} (8.1)

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

(ab)c=abc\left(a^{b}\right)^{c}=a^{bc} (8.2)

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