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Difficulty: HardOrder of Operations and Number Properties

Four mathematical expressions are shown below:

* Expression 1: −32−12÷3×2-3^2 - 12 \div 3 \times 2
* Expression 2: −∣−4−2×3∣+(−2)3-| -4 - 2 \times 3 | + (-2)^3
* Expression 3: 4−3(2−32)4 - 3(2 - 3^2)
* Expression 4: −24÷8×(−1−1)2-2^4 \div 8 \times (-1 - 1)^2

Order the expressions by their simplified numerical values from least to greatest.

  1. 1Expression 2
  2. 2Expression 1
  3. 3Expression 4
  4. 4Expression 3

Answer

The correct order of the expressions from least to greatest value is Expression 2, Expression 1, Expression 4, and Expression 3.
Evaluating each expression using the correct order of operations yields: Expression 1 equals −17-17, Expression 2 equals −18-18, Expression 3 equals 2525, and Expression 4 equals −8-8. Arranging these values from least to greatest results in −18<−17<−8<25-18 < -17 < -8 < 25, corresponding to the sequence: Expression 2, Expression 1, Expression 4, Expression 3.

Step-by-Step Solution

1
Evaluate Expression 1.
−17-17
Apply the order of operations: first evaluate the exponent −32=−9-3^2 = -9. Next, perform multiplication and division from left to right: 12÷3=412 \div 3 = 4, and 4×2=84 \times 2 = 8. Finally, perform the subtraction: −9−8=−17-9 - 8 = -17.
2
Evaluate Expression 2.
−18-18
Simplify inside the absolute value first by performing multiplication before subtraction: 2×3=62 \times 3 = 6, and −4−6=−10-4 - 6 = -10. The absolute value of −10-10 is 1010, which becomes −10-10 due to the negative sign outside. Next, evaluate the exponent: (−2)3=−8(-2)^3 = -8. Finally, add the terms: −10+(−8)=−18-10 + (-8) = -18.
3
Evaluate Expression 3.
25
Simplify inside the parentheses first, evaluating the exponent before subtracting: 32=93^2 = 9, and 2−9=−72 - 9 = -7. Next, perform the multiplication: 3×(−7)=−213 \times (-7) = -21. Finally, subtract: 4−(−21)=4+21=254 - (-21) = 4 + 21 = 25.
4
Evaluate Expression 4.
−8-8
Simplify the expression inside the parentheses: −1−1=−2-1 - 1 = -2. Next, evaluate the exponents: −24=−16-2^4 = -16 and (−2)2=4(-2)^2 = 4. Perform division and multiplication from left to right: −16÷8=−2-16 \div 8 = -2, and −2×4=−8-2 \times 4 = -8.
5
Compare the simplified values to order them from least to greatest.
−18<−17<−8<25-18 < -17 < -8 < 25
Comparing the results shows that the values in ascending order are −18-18 (Expression 2), −17-17 (Expression 1), −8-8 (Expression 4), and 2525 (Expression 3).

Key Concept

Order of Operations and Number Properties
Estimated Time:2m 30s
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Topics

Order of Operations and Number Properties

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