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Difficulty: MediumOrder of Operations and Number Properties

Arrange the following mathematical expressions in order of their simplified values from least to greatest.

  1. 1−3−(4−2) -3 - (4 - 2)
  2. 2−32+4×2 -3^2 + 4 \times 2
  3. 3(−3)2−4×2 (-3)^2 - 4 \times 2
  4. 4−(3−4×2) -(3 - 4 \times 2)

Answer

The correct order of the expressions from least to greatest simplified value is: −3−(4−2)-3 - (4 - 2), then −32+4×2-3^2 + 4 \times 2, then (−3)2−4×2(-3)^2 - 4 \times 2, and finally −(3−4×2)-(3 - 4 \times 2).
Evaluating each expression using the correct order of operations yields −5-5 for −3−(4−2)-3 - (4 - 2), −1-1 for −32+4×2-3^2 + 4 \times 2, 11 for (−3)2−4×2(-3)^2 - 4 \times 2, and 55 for −(3−4×2)-(3 - 4 \times 2). Ordering these results from least to greatest yields the sequence: −3−(4−2)-3 - (4 - 2), followed by −32+4×2-3^2 + 4 \times 2, then (−3)2−4×2(-3)^2 - 4 \times 2, and finally −(3−4×2)-(3 - 4 \times 2).

Step-by-Step Solution

1
Simplify the expression −3−(4−2)-3 - (4 - 2).
−3−(4−2)=−3−2=−5-3 - (4 - 2) = -3 - 2 = -5
Perform the operation inside the parentheses first, then subtract the result from the leading term.
2
Simplify the expression −32+4×2-3^2 + 4 \times 2.
−32+4×2=−9+8=−1-3^2 + 4 \times 2 = -9 + 8 = -1
Evaluate the exponent first, noting that the negative sign is outside the base (−32=−9 -3^2 = -9 ), then multiply, and finally add the terms.
3
Simplify the expression (−3)2−4×2(-3)^2 - 4 \times 2.
(−3)2−4×2=9−8=1(-3)^2 - 4 \times 2 = 9 - 8 = 1
Evaluate the term inside parentheses raised to the power first ((−3)2=9(-3)^2 = 9), then perform multiplication, and subtract.
4
Simplify the expression −(3−4×2)-(3 - 4 \times 2).
−(3−4×2)=−(3−8)=−(−5)=5-(3 - 4 \times 2) = -(3 - 8) = -(-5) = 5
Inside the parentheses, perform multiplication before subtraction, then apply the negative sign outside the parentheses.
5
Compare the simplified values of the four expressions.
−5<−1<1<5-5 < -1 < 1 < 5
Arrange the resulting values from least to greatest to determine the correct sequence.

Key Concept

Applying the correct order of operations (PEMDAS/GEMS) to evaluate numerical expressions, specifically distinguishing between −a2-a^2 and (−a)2(-a)^2.
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Topics

Order of Operations and Number Properties

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