product rule of exponents

The product rule says that to multiply two powers of the same base, you keep the base and add the exponents: b^m × b^n = b^(m+n). It works because multiplying lines up all the factors in a row, and you just count them.

Take 2^3 × 2^2. The first is three 2s, the second is two 2s; written out, that is 2 × 2 × 2 × 2 × 2, which is five 2s, namely 2^5. Adding the exponents 3 + 2 = 5 simply counts the total number of factors without writing them all out.

The rule applies only when the bases match. You cannot collapse 2^3 × 3^2 into a single power, because the factors are different numbers. And do not confuse this with raising a power to a power, where the exponents multiply instead of add.