PEAZIT Aha card
Go straight to resources ↓Powers: the exponent counts equal factors
2³ contains three factors of 2. Multiplying powers with the same base adds the numbers of factors.
| Notation | Expanded |
|---|---|
| 2² × 2³ | (2 × 2) × (2 × 2 × 2) |
| (2²)³ | (2 × 2) × (2 × 2) × (2 × 2) |
One worked example
- 2² × 2³Both powers have the same base, 2.
- 2 × 2 × 2 × 2 × 2Two factors of 2 plus three factors of 2 make five factors.
- 2⁵ = 32In general aᵐ × aⁿ = aᵐ⁺ⁿ for nonnegative integer exponents wherever the expression is defined.
Watch out: The product rule needs equal bases and multiplication. Do not add exponents in 2² + 2³. Zero and negative exponents require a nonzero base.
Goal, prerequisites and card origin
What you will learn: Distinguish a product of powers from a power of a power.
What you need first: Multiplication and brackets.
An original PEAZIT working card. Independent expert review and effectiveness testing have not yet taken place. · v1
OpenStax: exponent properties ↗Show connections
2³ contains three factors of 2. Multiplying powers with the same base adds the numbers of factors.