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Question

Difficulty: Very hardOrder of Operations

Given the variables x=−2x = -2, y=−3y = -3, and z=11z = 11, evaluate the algebraic expression below.

x3−2y(z−x)∣y2−z∣+x\frac{x^3 - 2y(z - x)}{|y^2 - z|} + x

Which of the following represents the correctly calculated result?

  1. A
    -45
  2. 33Answer
  3. C
    -15
  4. D
    -21

Answer

The correct result is 33.
The correct answer shows the result of correctly substituting the negative values, applying the order of operations by evaluating the innermost parentheses and exponents first, then multiplication, subtraction, and finally division and addition.

Step-by-Step Solution

1
Substitute the given values for x, y, and z into the expression.
(−2)3−2(−3)(11−(−2))∣(−3)2−11∣+(−2)\frac{(-2)^3 - 2(-3)(11 - (-2))}{|(-3)^2 - 11|} + (-2)
Accurate substitution, particularly retaining negative signs inside parentheses, is the required first step before performing arithmetic operations.
2
Evaluate the innermost grouping symbols and the exponents.
11−(−2)=1311 - (-2) = 13, (−2)3=−8(-2)^3 = -8, and (−3)2=9(-3)^2 = 9. The expression becomes −8−2(−3)(13)∣9−11∣−2\frac{-8 - 2(-3)(13)}{|9 - 11|} - 2.
Following the order of operations, computations inside parentheses and exponents must be resolved before multiplication.
3
Perform the multiplication in the numerator and the subtraction within the absolute value in the denominator.
Numerator multiplication: 2(−3)(13)=−6(13)=−782(-3)(13) = -6(13) = -78. Absolute value: ∣9−11∣=∣−2∣=2|9 - 11| = |-2| = 2. The expression is now −8−(−78)2−2\frac{-8 - (-78)}{2} - 2.
Multiplication precedes addition and subtraction. The absolute value acts as a grouping symbol that must be evaluated before division.
4
Simplify the numerator by resolving the subtraction of a negative number.
−8−(−78)=−8+78=70-8 - (-78) = -8 + 78 = 70. The expression simplifies to 702−2\frac{70}{2} - 2.
Subtracting a negative number is mathematically equivalent to adding its positive counterpart.
5
Perform the division, then the final subtraction.
702=35\frac{70}{2} = 35, and 35−2=3335 - 2 = 33.
Division must be completed before the final addition or subtraction step.

Key Concept

Evaluating complex algebraic expressions with negative numbers using the correct order of operations.
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Topics

Order of Operations

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