Laws of exponentsMultiplying powers
Name: Date:
Write each product as one power. Give the exponent n.
13⁵ × 3⁸ = 3n, n =
27¹² × 7⁵ = 7n, n =
38¹⁵ × 8³ = 8n, n =
412³ × 12¹² = 12n, n =
54² × 4⁹ = 4n, n =
63¹¹ × 3⁴ = 3n, n =
Product rule for exponents worksheets
Multiply two powers of the same base and the exponents add: that one move is the first rule of exponents a course teaches. These free printable worksheets ask for the exponent alone, so every answer is a whole number.
Kid stuck on one? You don’t just get the sheet.
Product rule for exponents
The answer key prints on a separate page.
What's on this sheet
2³ × 2⁵ is three 2s multiplied by five 2s, which is eight 2s in a row — so the answer is 2⁸, and the exponent is simply 3 + 5. Writing it as 2ⁿ rather than as 32768 is what keeps the question about the rule: a child who works out both values and multiplies them has done real arithmetic but has never met the pattern the standard is after.
Why this sheet works
The base has to be the same for the exponents to add, and the sheet holds a child to that. 2³ × 2⁵ combines because both are powers of 2; 2³ × 5² does not, because there is no single base to count. Keeping to one base per question is what makes "add the exponents" a rule rather than a coincidence.
How this one works
One question from this sheet, worked through a step at a time.
3⁵ × 3⁸
3⁵ × 3⁸
3 × 3 × 3 × 3 × 3 × 8 3s multiplied together
3⁵ × 3⁸=3¹³
n=13
- 3⁵ means 5 3s multiplied together, and 3⁸ means 8 3s multiplied together. The question multiplies those two lots together.
- So altogether that is 5 3s and 8 more — 5 + 8 = 13 of them.
- 13 3s multiplied together is 3¹³. So n = 13. That is the shortcut: multiplying powers of the same base ADDS the exponents.