Exponent rules

A power is shorthand for repeated multiplication. 2^3 is three 2s multiplied together, and 2^2 is two 2s multiplied together. When you multiply the two groups, you are multiplying five 2s in a row, and 5 is exactly 3 + 2, the sum of the exponents. Expand both powers first and the shortcut is just counting: three factors times two factors leaves five factors.

Rafi's mistake was multiplying the exponents, 3 * 2 = 6. That rule belongs to a different situation: 2^(3*2) would mean a power raised to a power, or (2^3)^2, which is a power of a power. Multiplying when you are multiplying the powers themselves adds a layer that is not in the problem, and his answer 2^6 = 64 is far too big.

The fix is to count the factors. 2^3 * 2^2 = (2 * 2 * 2) * (2 * 2) = 2^5 by length. Then evaluate: 2^5 = 32. Each doubled step, 2, 4, 8, 16, 32, confirms the count of five factors, and the fifth doubling lands on 32, exactly the product of 8 and 4. The doubling chain is its own built-in check, because each link must double.

Check with the expanded forms: 8 * 4 = 32 matches 2^5, so the rule gives the same number as counting. The same base is the key condition: this shortcut only works when the bases match, and the exponents add because the factors join in one long chain of the same repeated base. Different bases, and the shortcut sits out: the counting way still works.