The Rule in Letters

The power of a power rule written with letters is (a^m)^n = a^(mn). It holds for any base a and any exponents m and n. Each copy of a^m holds m factors of a, and there are n copies, so there are mn factors of a in all.

For example, with a = x, m = 3 and n = 2, the rule gives (x^3)^2 = x^6.

You now have three laws for exponents with the same base. The product rule, a^m * a^n = a^(m+n), is for two powers multiplied side by side, and the exponents add. The quotient rule, a^m / a^n = a^(m-n), is for one power divided by another, and the exponents subtract. The power of a power rule, (a^m)^n = a^(mn), is for a power inside a power, and the exponents multiply. All three come from counting factors of the base.