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How is the order of operations used to evaluate numerical and algebraic expressions?

Published in Order of Operations 2 mins read

The order of operations provides a standardized sequence for simplifying mathematical expressions to ensure everyone arrives at the same correct answer. This order is crucial for both numerical and algebraic expressions. According to established mathematical conventions, the correct sequence to follow is often remembered by the acronym PEMDAS (or sometimes BODMAS).

Understanding PEMDAS

PEMDAS dictates the order in which operations must be performed:

Order Operation Description
1 Parentheses First, simplify any expressions inside parentheses or brackets. This includes any operation within these grouping symbols.
2 Exponents Second, evaluate all exponents (powers and roots).
3 Multiplication Third, perform all multiplication and division operations from left to right. These operations have equal precedence.
4 Division
5 Addition Fourth, complete all addition and subtraction operations from left to right. These operations also have equal precedence.
6 Subtraction

Applying the Order of Operations

Here's how PEMDAS is applied in practice:

  1. Parentheses/Brackets: Start by simplifying anything inside parentheses, brackets, or other grouping symbols. This might involve applying PEMDAS within the parentheses themselves.

    • Example: 2 (3 + 4) = 2 7 = 14

  2. Exponents: Next, calculate any exponents present in the expression.

    • Example: 5 + 23 = 5 + 8 = 13

  3. Multiplication and Division: Perform multiplication and division from left to right. The order is determined by which operation appears first as you read from left to right.

    • Example: 10 / 2 3 = 5 3 = 15 (Division is performed before multiplication because it comes first from left to right)

  4. Addition and Subtraction: Finally, carry out addition and subtraction from left to right, similar to multiplication and division.

    • Example: 8 - 5 + 2 = 3 + 2 = 5 (Subtraction is performed before addition because it comes first from left to right)

Order of Operations in Algebraic Expressions

The same rules apply to algebraic expressions, where variables are involved.

  • Example: Simplify 3x + 2(y - 1) when x = 2 and y = 5

    1. Substitute the variable values: 3(2) + 2(5 - 1)
    2. Parentheses: 3(2) + 2(4)
    3. Multiplication: 6 + 8
    4. Addition: 14

Importance of Order

Following the order of operations ensures consistent and accurate results in mathematics, preventing ambiguity and errors in calculations.

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