Divide two powers of the same base and the exponents subtract — the companion to the product rule, and the one that leads somewhere new. These free printable worksheets ask for the exponent alone.
5⁷ ÷ 5² cancels two 5s from the seven on top, leaving five 5s — so the answer is 5⁵, and the exponent is 7 − 2. Asking for the exponent rather than the value is what makes the sheet about the rule: the numbers themselves get large fast, and a child who evaluates both and divides learns arithmetic but not the pattern.
Why this sheet works
This is the rule that reaches past whole numbers, and that is the reason to spend time on it. When the bottom exponent is larger the answer's exponent comes out negative, and when the two are equal it comes out zero — those are not awkward cases to avoid but exactly where negative and zero exponents come from. Cancel five 3s against seven and two are left underneath; cancel five against five and nothing is left but 1.
How this one works
One question from this sheet, worked through a step at a time.
10⁴ ÷ 10⁴
10 × 10 × 10 × 10 ÷ 10 × 10 × 10 × 10
10⁴ ÷ 10⁴=1
10⁴ ÷ 10⁴=10⁰=1
n=0
10⁴ is 4 10s on top and 10⁴ is 4 underneath. Dividing lets them cancel in pairs.
Every one of them cancels — the same 4 on top and underneath — so what is left is 1.
The rule says 4 − 4 = 0, so this is 10⁰. Both are right, which is exactly why 10⁰ = 1: it is what the rule has to mean. So n = 0.
