1Evaluating a Numerical Expression
What is the value of [(7² − 9) ÷ 8] × 2?
Answer Explanation
Work inside the parentheses first: 7² = 49, so 49 − 9 = 40.
Divide: 40 ÷ 8 = 5. Then multiply: 5 × 2 = 10.
PSAT/NMSQT Math Practice Problems: Exponents & Order of Operations
Order of operations questions on the PSAT/NMSQT Math section test whether you can work through a multi-step numerical expression — often stacked with parentheses, brackets, exponents, and fractions all at once — in exactly the right sequence: grouping symbols first (working from the innermost set outward), then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. A single misordered step early in a long expression cascades into a wrong final answer, even when every individual calculation along the way is done correctly.
The 8 problems below range from pure numerical order-of-operations expressions to multi-variable substitution problems where you first plug in given values and then simplify using the same careful step-by-step process. Every problem includes a complete step-by-step explanation, hidden by default so you can test yourself honestly before checking your work. For the more complex expressions, it helps to rewrite the expression after completing each individual step rather than trying to track multiple operations in your head at once — that's where most arithmetic slips happen. For unlimited additional practice, our free PSAT/NMSQT Math QBank has over 2,500 more problems across every PSAT/NMSQT Math topic, each with instant explanations.
2Evaluating a Numerical Expression with Brackets
What is the value of 19 − 3[20 − (2⁴−7)/4 × 8]?
Answer Explanation
Work inside the innermost parentheses first: 2⁴ = 16, so 16 − 7 = 9.
Continue inside the brackets: 9/4 = 2.25, then 2.25 × 8 = 18. So the bracket becomes 20 − 18 = 2. Finally: 19 − 3(2) = 19 − 6 = 13.
3Evaluating a Numerical Expression
What is the value of (72 ÷ 3² · 2) / 6?
Answer Explanation
Work the numerator first, left to right after handling the exponent: 3² = 9, so 72 ÷ 9 = 8, and 8 · 2 = 16.
Divide by the denominator: 16/6 = 8/3 (approximately 2.67).
4Evaluating a Numerical Expression
What is the value of 5³ − (1/2)(12 + 12 ÷ 3)?
Answer Explanation
Work inside the parentheses first, using order of operations within it: 12 ÷ 3 = 4, so 12 + 4 = 16.
Multiply: (1/2)(16) = 8. Then compute the exponent: 5³ = 125. Finally: 125 − 8 = 117.
5Evaluating an Expression with Substitution
What is the value of (2c/a)² − 10 × (b+a)/c if a = −2, b = 3, and c = 5?
Answer Explanation
Substitute and simplify the first term: 2c/a = 2(5)/(−2) = 10/−2 = −5, so (2c/a)² = (−5)² = 25.
Substitute and simplify the second term: (b+a)/c = (3 + (−2))/5 = 1/5, so 10 × (1/5) = 2. Finally: 25 − 2 = 23.
6Evaluating an Expression with Substitution
What is the value of 9 − 2x ÷ (z−y)³ if x = 4, y = −1, and z = −3?
Answer Explanation
Simplify inside the parentheses first: z − y = −3 − (−1) = −2, so (z−y)³ = (−2)³ = −8.
Compute the division: 2x ÷ (−8) = 8 ÷ (−8) = −1. Finally: 9 − (−1) = 10.
7Evaluating a Complex Fraction with Substitution
What is the value of [7 ÷ (q)² · 2]/(2p) · (−p+6q−r)/(−q) if p = 4, q = 1/2, and r = 2?
Answer Explanation
Simplify the first fraction: q² = (1/2)² = 1/4, so 7 ÷ (1/4) = 28, and 28 · 2 = 56. Dividing by 2p = 2(4) = 8 gives 56/8 = 7.
Simplify the second fraction: −p+6q−r = −4 + 6(0.5) − 2 = −4 + 3 − 2 = −3, and −q = −0.5, so −3/−0.5 = 6. Multiply the two results: 7 × 6 = 42.
8Evaluating a Complex Fraction with Substitution
What is the value of [c − 2(a+b)]/(c−a)² if a = −1/2, b = 3/2, and c = 5/2?
Answer Explanation
Simplify the numerator: a + b = −1/2 + 3/2 = 1, so 2(a+b) = 2, and c − 2(a+b) = 5/2 − 2 = 1/2.
Simplify the denominator: c − a = 5/2 − (−1/2) = 6/2 = 3, so (c−a)² = 9. Finally: (1/2)/9 = 1/18.
Keep building your order of operations skills
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