Exponents and radicals on the SAT: lesson and practice
With the same base, add exponents when multiplying, subtract when dividing, and multiply for a power of a power. A negative exponent means "one over" (x⁻² = 1/x²), and a fractional exponent is a root: xm/n = (ⁿ√x)m.
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The rules
| Rule | Example |
|---|---|
| xa · xb = xa+b | x³ · x⁴ = x⁷ |
| xa ÷ xb = xa−b | x⁸ ÷ x² = x⁶ |
| (xa)b = xab | (x³)⁴ = x¹² |
| x−a = 1/xa | 2⁻³ = 1/8 |
| xm/n = (ⁿ√x)m | 82/3 = (∛8)² = 4 |
Simplifying square roots
√50 + √18 = 5√2 + 3√2 = 8√2
Only add roots after simplifying them to the same radical. √50 + √18 is not √68.
Exponential equations
Rewrite both sides with the same base, then set the exponents equal: 9x = 27x−1 → 32x = 33x−3 → 2x = 3x − 3 → x = 3.
Common mistakes
Sample questions with answers
1. Which is equivalent to x³ · x⁵?
- x¹⁵
- x⁸ (correct)
- 2x⁸
- x²
Same base, multiplying: add the exponents, 3 + 5 = 8.
2. What is the value of 2⁻³?
- −8
- −1/8
- 1/8 (correct)
- 1/6
A negative exponent means one over: 1/2³ = 1/8.
3. Which is equivalent to √72?
- 36√2
- 6√2 (correct)
- 2√6
- 8√3
√72 = √36 × √2 = 6√2.
Frequently asked questions
What exponent rules are on the SAT?
Product (add exponents), quotient (subtract), power of a power (multiply), negative exponents (reciprocal) and fractional exponents (roots).
What does a fractional exponent mean?
The denominator is a root and the numerator is a power: 8^(2/3) is the cube root of 8, squared, which is 4.
How do you simplify a square root on the SAT?
Split the number into a perfect square times the rest, then take the square root of the perfect square: √72 = 6√2.
Can I use the calculator for exponent questions?
Yes. Desmos can evaluate powers and roots, and you can test equivalent expressions by substituting a value.