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Difficulty: HardOrder of Operations

An engineer uses the formula below to calculate a specific stress factor in a structural component:

p3−4pqr−q+(r−p)2 p^3 - \frac{4pq}{r - q} + (r - p)^2

If the component's parameters are measured as p=−2p = -2, q=−1q = -1, and r=3r = 3, what is the calculated stress factor?

  1. A
    -35
  2. 15Answer
  3. C
    -9
  4. D
    31

Answer

15
The correct answer is found by strictly following the order of operations (PEMDAS) while properly applying integer sign rules during substitution. First, evaluating grouping symbols yields a denominator of 4 and an exponent base of 5. Second, resolving the exponents gives -8 and 25. Third, calculating the middle fraction evaluates to 2. Finally, computing -8 - 2 + 25 from left to right results in 15.

Step-by-Step Solution

1
Substitute the given values into the expression.
(−2)3−4(−2)(−1)3−(−1)+(3−(−2))2(-2)^3 - \frac{4(-2)(-1)}{3 - (-1)} + (3 - (-2))^2
Variable substitution must be done precisely, maintaining all negative signs within parentheses to avoid sign errors.
2
Simplify the expressions acting as grouping symbols (the denominator and the base of the final exponent).
(−2)3−4(−2)(−1)4+(5)2(-2)^3 - \frac{4(-2)(-1)}{4} + (5)^2
According to the order of operations, operations inside parentheses and implied grouping symbols like fraction bars must be completed first.
3
Evaluate the exponents.
−8−4(−2)(−1)4+25-8 - \frac{4(-2)(-1)}{4} + 25
Exponents are the next priority in the order of operations. Note that an odd exponent preserves the negative sign of its base: (−2)3=−8(-2)^3 = -8.
4
Perform the multiplication in the numerator, then the division.
−8−84+25=−8−2+25-8 - \frac{8}{4} + 25 = -8 - 2 + 25
Multiplication and division are evaluated from left to right before addition and subtraction.
5
Perform the remaining addition and subtraction from left to right.
−10+25=15-10 + 25 = 15
Addition and subtraction have equal priority and must be evaluated strictly from left to right.

Key Concept

Evaluating Complex Expressions using Order of Operations and Negative Numbers
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Topics

Order of Operations

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