1.2 Order of Operations & Grouping Symbols

1.2 Order of Operations & Grouping Symbols

Mathematical expressions often involve multiple operations including addition, subtraction, multiplication, division, exponents, and grouping symbols. Without a standardized order of operations, an expression like $6 + 4 \times 5$ could yield two different answers: $50$ (if adding first) or $26$ (if multiplying first). To ensure unique, unambiguous evaluations across science, engineering, and mathematics, the standard Order of Operations rules are strictly enforced.

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The PEMDAS Hierarchy

The order of operations is governed by the widely recognized acronym PEMDAS:

1. P - Parentheses & Grouping Symbols: Evaluate all operations contained inside grouping symbols first, working from the innermost set outward.
2. E - Exponents & Roots: Evaluate all exponential powers and radical expressions next.
3. M & D - Multiplication & Division: Perform multiplication and division from left to right in the exact order they appear.
4. A & S - Addition & Subtraction: Perform addition and subtraction from left to right in the exact order they appear.

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Equal Precedence & Left-to-Right Execution

A critical rule tested heavily on the ACCUPLACER is that Multiplication does NOT take priority over Division, nor does Addition take priority over Subtraction.

• Multiplication and Division share identical rank. You execute whichever operation appears first as you read the expression from left to right.
• Addition and Subtraction share identical rank. You execute whichever operation appears first as you read the expression from left to right.

Example of Equal Precedence:

Consider $24 \div 4 \times 2$:

• Correct (Left-to-Right): $24 \div 4 = 6$, then $6 \times 2 = \mathbf{12}$.
• Incorrect (Forcing Multiplication First): $4 \times 2 = 8$, then $24 \div 8 = \mathbf{3}$ (Incorrect!).

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Nested Grouping Symbols & Fraction Bars

Expressions may contain multiple types of grouping symbols:

• Parentheses ()
• Brackets []
• Braces {}
• Fraction Bars $\frac{\text{Numerator}}{\text{Denominator}}$

Rule for Nested Symbols

Always isolate the innermost set of parentheses or brackets, simplify that expression to a single value, and then work outward step by step.

Fraction Bars as Implicit Grouping Symbols

A horizontal fraction bar acts as a grouping symbol around the entire numerator and the entire denominator: $\frac{A + B}{C + D} \equiv (A + B) \div (C + D)$ You must evaluate the entire numerator to a single number and the entire denominator to a single number before carrying out the final division.

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Step-by-Step Worked Examples

Example 1: Standard PEMDAS with Left-to-Right Rule

Problem: Evaluate $48 \div 6 \times 2 + 3^3 - 5$.

Step-by-Step Solution:

1. Parentheses: None present.
2. Exponents: Evaluate $3^3 = 27$. $\text{Expression: } 48 \div 6 \times 2 + 27 - 5$
3. Multiplication & Division (Left to Right):
   • First operation left-to-right is division: $48 \div 6 = 8$.
   • Next is multiplication: $8 \times 2 = 16$. $\text{Expression: } 16 + 27 - 5$
4. Addition & Subtraction (Left to Right):
   • Addition first: $16 + 27 = 43$.
   • Subtraction next: $43 - 5 = \mathbf{38}$.

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Example 2: Nested Brackets and Parentheses

Problem: Simplify $5 \times [40 - 2 \times (3^2 + 4)] + 18 \div 3$.

Step-by-Step Solution:

1. Innermost Parentheses $(3^2 + 4)$:
   • Exponent inside: $3^2 = 9$.
   • Addition inside: $9 + 4 = 13$. $\text{Expression: } 5 \times [40 - 2 \times 13] + 18 \div 3$
2. Outer Brackets $[40 - 2 \times 13]$:
   • Multiplication inside bracket takes priority over subtraction: $2 \times 13 = 26$.
   • Subtraction inside bracket: $40 - 26 = 14$. $\text{Expression: } 5 \times 14 + 18 \div 3$
3. Multiplication & Division (Left to Right):
   • Multiplication: $5 \times 14 = 70$.
   • Division: $18 \div 3 = 6$. $\text{Expression: } 70 + 6$
4. Final Addition: $70 + 6 = \mathbf{76}$.

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Example 3: Fraction Bar Grouping

Problem: Evaluate $\frac{6^2 - 4 \times (5 - 2)}{2^3 + 3 \times 4 - 14}$.

Step-by-Step Solution:

1. Numerator: $6^2 - 4 \times (5 - 2)$
   • Parentheses: $5 - 2 = 3 \implies 6^2 - 4 \times 3$
   • Exponent: $6^2 = 36 \implies 36 - 4 \times 3$
   • Multiplication: $4 \times 3 = 12 \implies 36 - 12 = \mathbf{24}$
2. Denominator: $2^3 + 3 \times 4 - 14$
   • Exponent: $2^3 = 8 \implies 8 + 3 \times 4 - 14$
   • Multiplication: $3 \times 4 = 12 \implies 8 + 12 - 14$
   • Left-to-right addition/subtraction: $8 + 12 = 20 \implies 20 - 14 = \mathbf{6}$
3. Final Division: $\frac{\text{Numerator}}{\text{Denominator}} = \frac{24}{6} = \mathbf{4}$

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ACCUPLACER Exam Traps & Common Errors