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Difficulty: MediumOrder of Operations

An engineer is using the formula P=2a3−b[a−(c−b)]2P = 2a^3 - b[a - (c - b)]^2 to calculate the tolerance of a specific component. If the parameters are measured as a=−2a = -2, b=3b = 3, and c=−1c = -1, what is the calculated value of PP?

  1. -28Answer
  2. B
    -52
  3. C
    4
  4. D
    -124

Answer

The calculated value of P is -28.
The correct answer is derived by substituting the values and strictly following PEMDAS: resolving the innermost parentheses to get -4, resolving the brackets to get 2, cubing -2 to get -8 and squaring 2 to get 4, performing the multiplications to yield -16 and 12, and subtracting to reach the final value of -28.

Step-by-Step Solution

1
Substitute the given values into the formula.
P=2(−2)3−3[−2−(−1−3)]2P = 2(-2)^3 - 3[-2 - (-1 - 3)]^2
Variable substitution must be done carefully, keeping negative numbers in parentheses to avoid sign drops.
2
Evaluate the innermost grouping symbols first.
(−1−3)=−4(-1 - 3) = -4
According to the order of operations, innermost parentheses must be simplified before outer brackets.
3
Simplify the expression inside the brackets by handling the double negative.
[−2−(−4)]=[−2+4]=2[-2 - (-4)] = [-2 + 4] = 2
Subtracting a negative number is equivalent to adding its positive counterpart.
4
Apply the exponents to their respective bases.
(−2)3=−8(-2)^3 = -8 and 22=42^2 = 4
Exponents must be evaluated before multiplication. A negative base raised to an odd power remains negative.
5
Perform the multiplication operations.
2(−8)=−162(-8) = -16 and 3(4)=123(4) = 12
Multiplication precedes subtraction in the order of operations.
6
Perform the final subtraction.
−16−12=−28-16 - 12 = -28
Subtraction is the final step after all grouping, exponents, and multiplication are resolved.

Key Concept

Order of Operations with Variable Substitution and Signed Integers
Estimated Time:45s
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Topics

Order of Operations

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