Questions: Integer Order of Operations

Question 1

What is the value of −5²?

• A. 25, because squaring a negative gives a positive
• B. −25, because exponentiation applies to 5 first, then the negation is applied
• C. −10, because −5² means −5 + (−5)
• D. 0, because −5 and 5 cancel each other

Answer: (B)

−5² means −(5²) = −(25) = −25. The exponent applies only to the base 5, not to the negative sign in front. Negation is treated as multiplication by −1, which has lower precedence than exponentiation in the order of operations. If you wanted to square the entire quantity including the negative, you would write (−5)² = (−5)(−5) = 25. The distinction between −5² and (−5)² is one of the most consequential single-character differences in arithmetic.

Question 2

A student evaluates 8 − (−4 + 6) and gets 18. Which mistake explains this error?

• A. They evaluated addition before subtraction even though subtraction comes first reading left to right
• B. They distributed the subtraction incorrectly — treating −(−4 + 6) as +(−4) + 6 instead of subtracting the entire quantity (which equals 2)
• C. They computed −4 + 6 as −2 instead of +2
• D. They forgot to apply parentheses before subtraction

Answer: (B)

The correct evaluation: −4 + 6 = 2, then 8 − 2 = 6. The student got 18 by treating 8 − (−4 + 6) as 8 − (−4) + 6 = 8 + 4 + 6 = 18. This error comes from incorrectly distributing the subtraction: they flipped the sign of −4 to get +4, but then also kept the +6 instead of negating it. When subtracting a parenthetical expression, the subtraction applies to the entire result inside — not to each term separately unless you distribute a negative sign across all terms, which would give 8 + 4 − 6 = 6.

Question 3

−3² and (−3)² evaluate to the same number because both expressions involve squaring something negative.

• T. True
• F. False

Answer: False

−3² = −(3²) = −9, while (−3)² = (−3)(−3) = 9. They differ by a sign of 18. The difference is what is being squared: in −3², only the 3 is squared and the negative is applied afterward. In (−3)², the entire quantity −3 is squared. The parentheses literally change the base of the exponent from 3 to −3, which changes the result entirely. This is one of the most important distinctions in integer arithmetic.

Question 4

In the expression −2 · (−3) · (−4), the product is negative because there is an odd number of negative factors.

• T. True
• F. False

Answer: True

Counting negative factors is a reliable shortcut for determining the sign of a product or quotient: an even count of negative factors gives a positive result; an odd count gives a negative result. Here there are three negative factors (−2, −3, −4), which is odd, so the product is negative. Checking: (−2)(−3) = 6, then 6 · (−4) = −24. The shortcut works because each pair of negatives cancels to a positive, and one negative is left over.

Question 5

Explain why −3² equals −9 rather than 9. What order-of-operations rule governs this, and how does writing (−3)² instead change the result?

Model answer: In −3², the order of operations requires exponentiation before any multiplication (including the implicit multiplication by −1 that represents negation). So the 3 is squared first to get 9, and then the negative sign is applied: −(9) = −9. In (−3)², the parentheses change what is being squared — now the entire quantity −3 is the base of the exponent, so (−3)² = (−3)(−3) = 9. The parentheses move the negative sign inside the base, so it participates in the squaring rather than being applied afterward.

The core insight is that negation is not the same as being part of the base. Without parentheses, the negative sign is an operation applied to the result of the exponentiation, not a property of the number being raised to a power. This is why −x² is always non-positive (for real x), while (−x)² = x² is always non-negative — a distinction with major consequences in algebra.