Just like we can define multiplication as repeated addition, we can define exponentiation as repeated multiplication. Exponentiation doesn't have the same properties as addition and multiplication, but it does have some worth mentioning, which again we can prove are true based on the definition. It does have closure on the natural numbers, so an is always a natural number if a and n are natural numbers. It is not commutative or associative (in general, an≠na), but it is true that am+n=aman, which feels to me like the distributive law for multiplication and addition. Also, (am)n=amn, which follows from the previous property.
Another property the natural numbers have is that they are ordered. a>b is defined to mean that there exists a natural number n such that a=b+n. This means that for any two numbers, there are three possibilities: they are equal, the first is greater, or the second is greater. The fact that one and only one of these three cases is true is used all the time in proofs. It also means that if a>b and b>c, a must be greater than c. Which means that for any set of natural numbers, it is always possible to order them from smallest to largest.
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