🔑 The 3 Big Rules to Remember
Multiply same base → ADD exponents | 2³ × 2⁴ = 2⁷
Divide same base → SUBTRACT exponents | 5⁶ ÷ 5² = 5⁴
Power of a power → MULTIPLY exponents | (4²)³ = 4⁶
bⁿ means multiply b by itself n times.
The base is the number being multiplied. The exponent is how many times.
In 2³: the base = 2, the exponent = 3.
2 × 2 × 2 = 8
What is 3⁴? (Multiply 3 by itself 4 times: 3×3×3×3 = ?)
b⁰ = 1 for any nonzero value of b.
This works because dividing a number by itself = 1: bⁿ ÷ bⁿ = b⁰ = 1.
Think about dividing: 5² ÷ 5² = 25 ÷ 25 = 1. Using the division rule (subtract exponents): 5²⁻² = 5⁰ = 1. They match!
100⁰ = 1 | 999⁰ = 1 | (−7)⁰ = 1 | (1/2)⁰ = 1
Note: 0⁰ is undefined.
What is 247⁰? What about (−15)⁰?
b⁻ⁿ = 1 / bⁿ
A negative exponent does NOT make the answer negative. It means take the reciprocal (1 over the base to the positive exponent).
3⁻² → write it as 1/3². The negative sign moves the base to the denominator.
3² = 3 × 3 = 9, so 1/3² = 1/9
⚠️ Common Mistake
3⁻² does NOT equal −9. The negative is in the exponent, not the answer. Always flip first, then calculate!
What is 2⁻³? (Flip: 1/2³ = 1/? )
bᵐ × bⁿ = bᵐ⁺ⁿ
When multiplying two powers with the same base, keep the base and add the exponents.
Both have base 2. ✅ Same base — we can add exponents.
3 + 4 = 7 → result is 2⁷
2⁷ = 2×2×2×2×2×2×2 = 128
What is 3² × 3³? (Add the exponents: 2+3=5, so 3⁵ = ?)
bᵐ ÷ bⁿ = bᵐ⁻ⁿ
When dividing two powers with the same base, keep the base and subtract the bottom exponent from the top.
Both have base 5. ✅ Same base — we can subtract exponents.
6 − 2 = 4 → result is 5⁴
5⁴ = 5×5×5×5 = 625
What is 4⁵ ÷ 4²? (Subtract exponents: 5−2=3, so 4³ = ?)
(bᵐ)ⁿ = bᵐˣⁿ
When raising a power to another power, keep the base and multiply the exponents.
(4²)³ — inner exponent = 2, outer exponent = 3, base = 4.
2 × 3 = 6 → result is 4⁶
4⁶ = 4,096
What is (2³)²? (Multiply exponents: 3×2=6, so 2⁶ = ?)
b^(1/2) = √b (square root)
b^(1/3) = ∛b (cube root)
b^(1/n) = ⁿ√b (nth root)
The denominator of the fractional exponent becomes the index of the root!
What is 64^(1/2)? What is 8^(1/3)?
| Law Name | Rule | Example |
|---|---|---|
| Positive Exponent | bⁿ = b×b×…×b (n times) | 2³ = 8 |
| Zero Exponent | b⁰ = 1 | 5⁰ = 1 |
| Negative Exponent | b⁻ⁿ = 1/bⁿ | 3⁻² = 1/9 |
| Product of Powers | bᵐ × bⁿ = bᵐ⁺ⁿ | 2³×2⁴ = 2⁷ = 128 |
| Quotient of Powers | bᵐ ÷ bⁿ = bᵐ⁻ⁿ | 5⁶÷5² = 5⁴ = 625 |
| Power of a Power | (bᵐ)ⁿ = bᵐˣⁿ | (4²)³ = 4⁶ = 4096 |
| Fractional (½) | b^(1/2) = √b | 16^(1/2) = 4 |
| Fractional (⅓) | b^(1/3) = ∛b | 27^(1/3) = 3 |
🔑 3-Word Memory Trick
Multiply → ADD (bᵐ × bⁿ = bᵐ⁺ⁿ)
Divide → SUBTRACT (bᵐ ÷ bⁿ = bᵐ⁻ⁿ)
Power → MULTIPLY ((bᵐ)ⁿ = bᵐˣⁿ)