Powers
A power like 2⁵ has a base (2) and an exponent (5): 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.
The exponent laws are shortcuts that follow directly from this meaning.
The laws
| Law | Rule | Example |
|---|---|---|
| Product law | xᵃ × xᵇ = xᵃ⁺ᵇ | x³ × x⁴ = x⁷ |
| Quotient law | xᵃ ÷ xᵇ = xᵃ⁻ᵇ | x⁸ ÷ x³ = x⁵ |
| Power of a power | (xᵃ)ᵇ = xᵃᵇ | (x²)⁵ = x¹⁰ |
| Power of a product | (xy)ᵃ = xᵃyᵃ | (2x)³ = 8x³ |
| Power of a quotient | (x/y)ᵃ = xᵃ/yᵃ | (x/3)² = x²/9 |
| Zero exponent | x⁰ = 1 (x ≠ 0) | 7⁰ = 1 |
Why they work
- x³ × x⁴ = (x·x·x)(x·x·x·x) = seven x’s → x⁷
- x⁵ ÷ x² = (x·x·x·x·x)/(x·x) → three x’s left → x³
- (x²)³ = x² · x² · x² = x⁶
- x³ ÷ x³ = 1, and by the quotient law it’s x⁰, so x⁰ = 1.
Same base only
The product and quotient laws need the same base.
- 2³ × 2⁴ = 2⁷ ✓
- 2³ × 3⁴ can’t be combined into one power.
With coefficients
Multiply or divide coefficients; apply the laws to variables.
- (3x²)(4x⁵) = 12x⁷
- (−2a³b)² = 4a⁶b²
- 15m⁶n² ÷ 5m²n = 3m⁴n
Negative bases
- (−3)² = 9 but −3² = −9
- (−2)³ = −8
Brackets matter!
Order of operations
Exponents come right after brackets: 2 × 3² = 2 × 9 = 18, not 6² = 36.
Combining laws
Example: (2x³)² × x⁴ ÷ (4x⁵)
- (2x³)² = 4x⁶
- 4x⁶ × x⁴ = 4x¹⁰
- 4x¹⁰ ÷ 4x⁵ = x⁵
Worked examples
Example 1: a⁴ × a⁹ → a¹³.
Example 2: (y³)⁴ → y¹².
Example 3: (5x²y)⁰ → 1.
Common mistakes
- Multiplying exponents in the product law: x² × x³ = x⁵, not x⁶.
- Multiplying the base: 2³ × 2⁴ ≠ 4⁷.
- (x + y)² = x² + y²: exponent laws don’t apply over addition.
- Forgetting to raise the coefficient: (3x)² = 9x², not 3x².
Try it yourself
- x⁵ × x²
- m¹⁰ ÷ m⁴
- (2a⁴)³
- (x²y³)²
- 12x⁷ ÷ (3x²)
Answers: 1) x⁷ 2) m⁶ 3) 8a¹² 4) x⁴y⁶ 5) 4x⁵
Quick recap
- Product law: add exponents. Quotient law: subtract them.
- Power of a power: multiply exponents.
- Zero exponent: x⁰ = 1.