ACT Math: Negative Numbers and Order of Operations

Students often mishandle negative signs because they confuse where the negative applies. The key is this: A negative sign immediately before a number or variable is part of that number. A negative sign in front of an exponent means "take the opposite of the result, not include the negative in the exponent." Example: -3^2 means -(3^2)=-(9)=-9, not (-3)^2=9. The parentheses make all the difference. Another critical case: (-2)^3=-8 (negative times negative times negative equals negative), but -2^3=-(8)=-8 anyway because the exponent applies only to the 2, and then the negative sign outside negates the result. Always identify whether the negative is inside or outside the parentheses/exponent before computing.

Practice: Evaluate -4^2 and (-4)^2. First: -(4^2)=-(16)=-16. Second: (-4)(-4)=16. Completely different answers. On the ACT, if these are answer choices and you misread the notation, you pick wrong. Read symbols precisely.

Trap 1: Forgetting that two negatives make a positive in multiplication or division. -8/-2=4 (both negatives cancel). Many students write -4 instead. Trap 2: Misplacing the negative under a square root. sqrt(-4) is not real (no real square root of negative), but -sqrt(4)=-2 (square root of 4, then negate). These look similar but are completely different. Trap 3: Dropping the negative when performing order of operations. "Simplify: -3+5×2" becomes -3+10=7, not -30. Multiplication happens first, then addition/subtraction left to right. The negative applies only to the 3, not to the whole expression. Write out each step explicitly until you're confident you're tracking negatives correctly.

Defensive move: Circle every negative sign you see and write what it's attached to. If a negative applies to 3, write "-3" in a box. If it applies to the whole result, note it separately.

Expression 1: -2^3=-8 (negative outside; cube of 2 is 8, then negate). Expression 2: (-2)^3=-8 (negative inside; -2 times -2 times -2 equals -8). Expression 3: -3^2+5=-(9)+5=-9+5=-4. Expression 4: 12/(-3)=-4 (positive divided by negative equals negative). Expression 5: -sqrt(9)=-3 (square root of 9 is 3, then negate). Check each answer by working backward: If the result is -4, what original expression gives that?

If you missed any, write out the step-by-step logic until you see your error. Most mistakes come from misreading symbols or skipping the order of operations.

Negative number and exponent questions appear on every ACT Math test, often disguised in larger problems. Students who handle negatives confidently can move through these questions in 30 seconds; those who second-guess themselves waste time and often get them wrong. One confident mastery of negative-number rules can save 3-5 minutes per test, which compounds into a higher overall score.

This week, drill negative and exponent problems separately from harder algebra. Build your confidence in the fundamentals, then apply them in context.