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Question

Difficulty: HardOrder of Operations and Number Properties

Evaluate the four mathematical expressions below and arrange them in order from least to greatest value.

  1. 1Expression X: −(−3)2−∣−6×2+5∣-(-3)^2 - | -6 \times 2 + 5 |
  2. 2Expression W: −24−3(5−7)2÷(−2)-2^4 - 3(5 - 7)^2 \div (-2)
  3. 3Expression Z: −18÷(−3)×(−2)−4+23÷4\frac{-18 \div (-3) \times (-2)}{-4 + 2^3 \div 4}
  4. 4Expression Y: 4−3×[8−(2−5)2]4 - 3 \times [8 - (2 - 5)^2]

Answer

Expression X, Expression W, Expression Z, Expression Y
Evaluating all four expressions using the proper order of operations (PEMDAS) yields the values: Expression X is −16-16, Expression W is −10-10, Expression Z is 66, and Expression Y is 77. Ordering these values from least to greatest results in the sequence Expression X, Expression W, Expression Z, Expression Y.

Step-by-Step Solution

1
Evaluate Expression X: −(−3)2−∣−6×2+5∣-(-3)^2 - | -6 \times 2 + 5 |
−16-16
First, evaluate the exponent: (−3)2=9(-3)^2 = 9, which makes the first term −9-9. Next, calculate the expression inside the absolute value bars: −6×2+5=−12+5=−7-6 \times 2 + 5 = -12 + 5 = -7. The absolute value of −7-7 is 77. Subtracting this value gives −9−7=−16-9 - 7 = -16.
2
Evaluate Expression W: −24−3(5−7)2÷(−2)-2^4 - 3(5 - 7)^2 \div (-2)
−10-10
First, evaluate the exponent: −24=−(24)=−16-2^4 = -(2^4) = -16. Then, evaluate inside the parentheses: 5−7=−25 - 7 = -2, and square it: (−2)2=4(-2)^2 = 4. Next, perform multiplication and division from left to right: −3(4)÷(−2)=−12÷(−2)=6-3(4) \div (-2) = -12 \div (-2) = 6. Adding this to −16-16 gives −16+6=−10-16 + 6 = -10.
3
Evaluate Expression Z: −18÷(−3)×(−2)−4+23÷4\frac{-18 \div (-3) \times (-2)}{-4 + 2^3 \div 4}
66
In the numerator, perform multiplication and division from left to right: −18÷(−3)=6-18 \div (-3) = 6, and then 6×(−2)=−126 \times (-2) = -12. In the denominator, evaluate the exponent: 23=82^3 = 8, divide: 8÷4=28 \div 4 = 2, and add: −4+2=−2-4 + 2 = -2. Finally, divide the numerator by the denominator: −12−2=6\frac{-12}{-2} = 6.
4
Evaluate Expression Y: 4−3×[8−(2−5)2]4 - 3 \times [8 - (2 - 5)^2]
77
Evaluate the innermost parentheses first: 2−5=−32 - 5 = -3, and square it: (−3)2=9(-3)^2 = 9. Simplify inside the brackets: 8−9=−18 - 9 = -1. Multiply: −3×(−1)=3-3 \times (-1) = 3. Add to the remaining term: 4+3=74 + 3 = 7.
5
Compare the evaluated values to find the correct ordering.
Expression X (−16-16) < Expression W (−10-10) < Expression Z (66) < Expression Y (77)
Comparing the values −16-16, −10-10, 66, and 77 from least to greatest establishes the correct sequence.

Key Concept

Order of operations dictates evaluating parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). Negation outside exponents and absolute values must be treated with care.
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Topics

Order of Operations and Number Properties

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