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Difficulty: Very hardOrder of Operations

Arrange the following calculation steps in the exact chronological sequence they must be performed relative to each other when evaluating the mathematical expression below using the standard order of operations:

−2[(−3)2−4(5−7)3]∣−9∣−(8−1)−9 \frac{-2[(-3)^2 - 4(5 - 7)^3]}{|-9| - (8 - 1)} - 9
  1. 1Evaluating the exponent expression \((5 - 7)^3\)
  2. 2Subtracting the product of \(4\) and \(-8\) from \(9\) within the brackets
  3. 3Multiplying the fully simplified value of the square brackets by \(-2\)
  4. 4Dividing the fully evaluated numerator by the fully evaluated denominator
  5. 5Subtracting \(9\) from the quotient

Answer

The correct sequence starts with the innermost exponents and grouping symbols, resolves the bracket, evaluates the full numerator and denominator, divides, and finally subtracts 9.
The correct order strictly follows the standard order of operations (PEMDAS). Operations within grouping symbols (innermost parentheses first, then brackets) must be evaluated before external operations. Exponents are applied immediately after their base is simplified. The fraction bar acts as a major grouping symbol, requiring the numerator and denominator to be fully evaluated before dividing. The final subtraction occurs last.

Step-by-Step Solution

1
Simplify the innermost parentheses and absolute value
\(5 - 7 = -2\) and \(8 - 1 = 7\) and \(|-9| = 9\)
Parentheses and absolute value brackets (P in PEMDAS) are evaluated first.
2
Evaluate the exponents
\((-3)^2 = 9\) and \((-2)^3 = -8\)
Exponents (E in PEMDAS) follow parentheses.
3
Perform multiplication inside the square brackets
\(4 \times (-8) = -32\)
Multiplication (M in PEMDAS) occurs before subtraction inside the grouping symbols.
4
Perform subtraction inside the square brackets
\(9 - (-32) = 41\)
Subtraction is the final operation needed to simplify the inside of the square bracket.
5
Multiply the bracket by the external coefficient
\(-2 \times 41 = -82\)
This completes the evaluation of the numerator.
6
Simplify the denominator
\(9 - 7 = 2\)
The denominator must be evaluated to a single number before division can occur.
7
Divide the numerator by the denominator
\(-82 \div 2 = -41\)
The fraction bar represents division of the evaluated top and bottom parts.
8
Perform the final subtraction
\(-41 - 9 = -50\)
Subtraction outside the fraction is the last operational step.

Key Concept

Order of Operations with Nested Grouping Symbols and Fractions
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Topics

Order of Operations

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