Exponents • Topic 1 of 4

Laws of Exponents

The exponent laws let you combine powers of the same base. Multiplying adds exponents (xᵃ·xᵇ = xᵃ⁺ᵇ); dividing subtracts them (xᵃ/xᵇ = xᵃ⁻ᵇ); raising a power to a power multiplies them ((xᵃ)ᵇ = xᵃᵇ); a power of a product distributes ((xy)ⁿ = xⁿyⁿ); and any nonzero base to the zero power is 1. These rules only combine like bases, so xᵃ·yᵇ does not simplify. Applying them in the right order keeps SAT exponent problems quick and avoids the classic error of adding exponents when you should multiply.

✅ Solved examples

1. Simplify x³ · x⁴.
Add the exponents: x³⁺⁴ = x⁷.
2. Simplify x⁸ / x³.
Subtract the exponents: x⁸⁻³ = x⁵.
3. Simplify (x²)⁵.
Multiply the exponents: x²ˣ⁵ = x¹⁰.
4. Evaluate 7⁰.
Any nonzero base to the zero power is 1.

✏️ Practice — try these, take hints as needed

1. Simplify x⁵ · x².
Same base → add exponents.
5 + 2.
—
x⁷.
2. Simplify x⁹ / x⁴.
Same base → subtract exponents.
9 − 4.
—
x⁵.
3. Simplify (x³)⁴.
Power of a power → multiply.
3 × 4.
—
x¹².
4. Simplify (2x)³.
Distribute the exponent.
2³ · x³.
—
8x³.
5. Simplify x⁶ · x⁰.
x⁰ = 1.
x⁶ · 1.
—
x⁶.

📝 Topic test — 8 questions

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