# What is an exponential object?

One interpretation for a power of natural numbers, such as $10^6$, is that it is counting a certain collection of functions: in this case, the number of functions from a six-element set to a ten-element set. (One way to think about it is that $10^6$ is the number of six-digit combinations from 000000 to 999999, and each combination corresponds to a function from $\{1,\ldots,6\}$ to $\{0,\dots,9\}$: the first digit is the image of $1$, the second digit is the image of $2$, etc.)

In general, the number $m^n$ is the number of functions from an $n$-element set to a $m$-element set. This is somewhat analogous to standard interpretations of other arithmetical operations:

Today we’ll explore these ideas for exponentiation instead of multiplication, and the story has a twist that finally explains why $(a^2)^{1/2}$ doesn’t always equal $a$.