Laws of exponentsA power of a power
Name: Date:
Write each power of a power as one power. Give the exponent n.
1(9³)⁵ = 9n, n =
2(7²)⁶ = 7n, n =
3(3⁵)⁷ = 3n, n =
4(8⁴)⁸ = 8n, n =
5(3⁴)⁸ = 3n, n =
6(4¹)² = 4n, n =
Power of a power worksheets
Raise a power to another power and the exponents multiply: (2³)⁴ is 2¹². These free printable worksheets practice that one rule, asking for the exponent alone so every answer is a whole number.
Kid stuck on one? You don’t just get the sheet.
Power of a power
The answer key prints on a separate page.
What's on this sheet
(2³)⁴ means four copies of 2³ multiplied together, and each 2³ is three 2s — so there are 4 × 3 = 12 twos in all, and the answer is 2¹². Written as 2ⁿ the question stays about the rule rather than the arithmetic, which for a value this size is the point.
Why this sheet works
The trap this sheet is built around is the one next door: here the exponents multiply, but when two powers are multiplied the exponents add. (2³)⁴ is 2¹², while 2³ × 2⁴ is 2⁷, and a child who has just practiced the product rule will often add these too. Reading the brackets as "this many copies of that power" is what tells the two rules apart.
How this one works
One question from this sheet, worked through a step at a time.
(9³)⁵
(9³)⁵
(9 × 9 × 9) × (9 × 9 × 9) × (9 × 9 × 9) × (9 × 9 × 9) × …
(9³)⁵=9¹⁵
n=15
- (9³)⁵ means 5 copies of 9³, all multiplied together.
- Each copy is 3 9s, and there are 5 copies — so that is 3 × 5 = 15 of them.
- 15 9s multiplied together is 9¹⁵. So n = 15. The shortcut: a power raised to a power MULTIPLIES the exponents.