Exponent rules and scientific notation are foundational skills tested throughout the PSAT/NMSQT Math section, often buried inside larger algebra problems rather than asked about directly. Being fast and accurate with the core exponent rules — multiplying powers by adding exponents, dividing powers by subtracting them, raising a power to a power by multiplying exponents, and handling negative and fractional exponents — pays off across nearly every topic on the test, not just questions that look like pure exponent drills.
The PSAT likes to combine several exponent rules in a single problem, often dressed up with multiple variables and coefficients at once. A typical question might ask you to simplify an expression like (−a²b³)(2ab²)(−3b) and then identify the exponents in the simplified form — testing whether you can track coefficients (multiplying the numbers separately from the variables) and exponents (adding them when the same base is multiplied) without mixing the two up. Scientific notation questions test a related but distinct skill: converting between standard form and scientific notation, and multiplying or dividing numbers written as a coefficient times a power of 10, where the powers of 10 combine using the same addition and subtraction rules as any other exponent.
Below are 9 PSAT/NMSQT-style practice problems covering exponent rules with multiple variables, negative and fractional exponents, and scientific notation calculations. Work through each problem, select your answer, and check whether you got it right — every problem includes a complete, step-by-step explanation. Once you've worked through these, keep building your skills with thousands more official-style questions in our free PSAT/NMSQT Math QBank.
If (−a²b³)(2ab²)(−3b) = kambn, what is the value of m + n?
Multiply the numerical coefficients separately from the variables: (−1)(2)(−3) = 6.
Multiply the a-terms by adding their exponents: a² × a¹ = a³ (m = 3).
Multiply the b-terms by adding their exponents: b³ × b² × b¹ = b⁶ (n = 6).
So the expression simplifies to 6a³b⁶, and m + n = 3 + 6 = 9. The correct answer is C.
If ((2/3)a²b)² ((4/3)ab)−3 = kambn, what is the value of k?
Apply the outer exponents to every factor inside each set of parentheses. For the first factor: ((2/3)a²b)² = (2/3)² a⁴ b² = (4/9)a⁴b².
For the second factor, a negative exponent means take the reciprocal and make the exponent positive: ((4/3)ab)⁻³ = (3/4)³ a⁻³ b⁻³ = (27/64)a⁻³b⁻³.
Multiply the two results together. Coefficients: (4/9)(27/64) = 108/576 = 3/16.
The value of k is 3/16 (the exponents work out to m = 1 and n = −1, but the question only asks for k). The correct answer is C.
If (x³(−y)²z⁻²) / (x⁻²y³z) = xm / (ynzp), what is the value of m + n + p?
Simplify the numerator first: since a negative number squared is positive, (−y)² = y², so the numerator is x³y²z⁻².
To divide powers of the same base, subtract the exponents (numerator minus denominator) for each variable separately:
x: 3 − (−2) = 5. y: 2 − 3 = −1. z: −2 − 1 = −3.
So the expression simplifies to x⁵y⁻¹z⁻³, which written with positive exponents in the given form x⁵/(y¹z³) gives m = 5, n = 1, p = 3. m + n + p = 5 + 1 + 3 = 9. The correct answer is C.
If 2x = 5, what is the value of 2x + 22x + 23x?
Rewrite each term using the power-of-a-power rule so everything is expressed in terms of 2ˣ: 2²ˣ = (2ˣ)² and 2³ˣ = (2ˣ)³.
Since 2ˣ = 5, substitute directly: 2ˣ + 2²ˣ + 2³ˣ = 5 + 5² + 5³.
= 5 + 25 + 125
= 155. The correct answer is C.
Which of the following is equivalent to the expression shown below?
Inside the parentheses, three identical terms are being added, not multiplied — adding identical terms is combining like terms, not an exponent rule: 3ˣ + 3ˣ + 3ˣ = 3 · 3ˣ.
Since 3 = 3¹, this becomes a multiplication of powers with the same base, so the exponents add: 3¹ · 3ˣ = 3¹⁺ˣ.
Now multiply by the remaining 3ˣ factor from outside the parentheses, again adding exponents: 3¹⁺ˣ · 3ˣ = 3¹⁺²ˣ.
The correct answer is D. A common mistake is multiplying the three terms inside the parentheses instead of adding them, which leads to choice B.
Keep the momentum going.
Explore 2,500+ Free PSAT/NMSQT Math ProblemsIf the expression below is written in the form axmyn, what is the value of m + n?
First expand (2xy)² = 4x²y², then multiply into the first factor: (6xy²)(4x²y²) = 24x³y⁴.
Now divide by the denominator, subtracting exponents for each matching base: coefficient 24/8 = 3; x³/x² = x¹; y⁴/y² = y².
The simplified expression is 3xy², so m = 1 and n = 2.
m + n = 1 + 2 = 3. The correct answer is B.
If x is not equal to zero, what is the value of the expression below?
Expand the numerator: (2x)³ = 8x³, so the numerator is 8x³ · 3x = 24x⁴.
Expand the denominator: (6x²)² = 36x⁴.
Divide: 24x⁴ / 36x⁴. Since x is not zero, the x⁴ terms cancel completely, leaving only the coefficients: 24/36 = 2/3.
The expression's value doesn't depend on x at all — it's simply 2/3. The correct answer is B.
If 8,200 × 300,000 is equal to 2.46 × 10n, what is the value of n?
Rewrite each number in scientific notation first: 8,200 = 8.2 × 10³ and 300,000 = 3 × 10⁵.
Multiply the coefficients and add the exponents on the powers of 10: (8.2 × 3) × 10³⁺⁵ = 24.6 × 10⁸.
This isn't yet in proper scientific notation, since 24.6 is not between 1 and 10. Move the decimal one place left and increase the exponent by 1: 24.6 × 10⁸ = 2.46 × 10⁹.
So n = 9. The correct answer is C.
If the expression below is equal to 1/(5 × 10n), what is the value of n?
Simplify each fraction separately first. 240/80,000 = 3/1,000. 6,000/900,000 = 1/150.
Multiply the simplified fractions: (3/1,000) × (1/150) = 3/150,000.
Simplify by dividing numerator and denominator by 3: 3/150,000 = 1/50,000.
Set this equal to the given form and solve for n: 1/50,000 = 1/(5 × 10n) → 5 × 10n = 50,000 → 10n = 10,000 → n = 4. The correct answer is B.
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