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Exponents for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT tests about exponents in one place: product and quotient of powers, power of a power, zero and negative exponents, and fractional exponents — with step-by-step examples, worked problems, and a free practice quiz.
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All exponent rules, negative exponents, and fractional exponents.
What is an exponent?
An exponent tells you how many times to multiply a base by itself: x⁴ means x · x · x · x. Exponent rules on the PSAT let you simplify and combine expressions without ever expanding them out by hand.
Every exponent rule below only combines terms that share the exact same base. Different bases can never be merged into a single exponent term.
Product & quotient of powers
Simplify: (x⁴ · x³) / x²
When multiplying powers with the same base, add the exponents — don't multiply them. x³ · x² = x⁵, not x⁶.
Power of a power & power of a product
Simplify: (2x³y²)²
When raising a power to another power, multiply the exponents. When raising an entire product to a power, distribute that exponent to every factor inside the parentheses, including any numeric coefficient.
Zero & negative exponents
Simplify: x³ / x⁷
A negative exponent does not make the value negative — it means "take the reciprocal." x⁻² equals 1/x², a positive fraction, not −x².
Fractional exponents
Evaluate: 27^(2/3)
In a fractional exponent m/n, the denominator (n) tells you which root to take, and the numerator (m) tells you what power to raise it to. Taking the root first, before applying the power, usually keeps the numbers smaller and easier to work with.
Simplifying expressions with exponents
Simplify: (x⁴y⁻²)³ / (x²y)
Common PSAT traps
- Multiplying instead of adding exponents: for xᴴ · xᴵ, add the exponents — multiplication of bases with the same exponent uses a different rule entirely.
- Combining different bases: exponent rules only work when the bases match exactly — x³ · y² cannot be simplified into a single power.
- Treating a negative exponent as a negative value: a negative exponent means "reciprocal," not "negative number."
- Forgetting to distribute an outer exponent to a coefficient: (3x)² = 9x², not 3x².
Test day strategy for exponents
| Question signal | Fastest approach |
|---|---|
| Multiplying same-base powers | Add the exponents |
| Dividing same-base powers | Subtract the exponents |
| A power raised to another power | Multiply the exponents |
| Negative exponent in a final answer | Rewrite as a reciprocal with a positive exponent |
| Fractional exponent to evaluate | Take the root indicated by the denominator, then apply the numerator's power |
Now put it to work
One comprehensive quiz covering the full topic — all exponent rules, negative exponents, and fractional exponents.
Exponents
All exponent rules, negative exponents, and fractional exponents.