Laws of Exponents
“Multiply and add exponents; divide and subtract them.”
The formula
aᵐ · aⁿ = aᵐ⁺ⁿHow to read it: multiplying powers with the same base adds the exponents
- aᵐ · aⁿ = aᵐ⁺ⁿ
- — multiply same bases → add the exponents
- (aᵐ)ⁿ = aᵐⁿ
- — a power of a power → multiply the exponents
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- — divide same bases → subtract the exponents
The hook
The laws of exponents look like a lot to memorize, but they're really one idea — counting repeated multiplication.
In plain words
A power is the same number multiplied several times. Multiply or divide those, and you just add or subtract the exponents.
The intuition
aᵐ is a multiplied m times, aⁿ is a multiplied n times. Multiply them and a is multiplied a total of m+n times, so the exponent is m+n. You're only counting 'how many times', which is why the exponents add.
How it's built
Three rules cover it all — multiply and add exponents (aᵐ·aⁿ=aᵐ⁺ⁿ), power a power and multiply exponents ((aᵐ)ⁿ=aᵐⁿ), divide and subtract exponents (aᵐ÷aⁿ=aᵐ⁻ⁿ).
Example
Write 2³ × 2² out fully: (2×2×2) × (2×2) is 2 five times. So 2⁵ = 32, exactly matching the exponent 3+2=5.
Common mistake
It's easy to mistake aᵐ · aⁿ for aᵐˣⁿ. Multiplication ADDS the exponents; you only MULTIPLY exponents when powering a power, as in (aᵐ)ⁿ.
Where it's used
Scientific notation for huge and tiny numbers (powers of 10), area and volume, and computer memory sizes in binary (2¹⁰=1024) — used every day.
Where it came from
The power notation aⁿ was popularized by Descartes, and the exponent laws were later extended naturally to rational, negative, and real exponents.
Prerequisites
Quick check
Write 2³ × 2⁴ as a single power.
- 2⁷✓
- 2¹²
- 2¹
- 4⁷
Practice
Compute 3² × 3³.
Answer: 243
- Add exponents: 3² × 3³ = 3⁵
- 3⁵ = 243
Takeaway: A product of the same base adds the exponents.
Compute (2³)².
Answer: 64
- Multiply exponents: (2³)² = 2⁶
- 2⁶ = 64
Takeaway: A power of a power multiplies the exponents.
Compute 5⁴ ÷ 5².
Answer: 25
- Subtract exponents: 5⁴ ÷ 5² = 5²
- 5² = 25
Takeaway: A quotient of the same base subtracts the exponents.
Write a⁵ · a³ as a single power.
Answer: undefined
- Same base, so add the exponents
- 5 + 3 = 8
- a⁵ · a³ = a⁸
Takeaway: The exponent laws only apply when the bases match.