Last modified on 21 November 2014, at 12:45

# Basic Algebra/Introduction to Basic Algebra Ideas/Order of Operations

## VocabularyEdit

Order of Operations
The order of operations is the rule at which you apply operations within a mathematical formula. There are two common mnemonics.

In mathematics, we use BODMAS:

• Brackets
• Orders (e.g. exponents)
• Division
• Multiplication
• Addition
• Subtraction

In the United States, you may also see PEMDAS:

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

Other variations exist, but the rules for order of operations remain the same.

## LessonEdit

Evaluate the expression $3 + 4 \times 5$.

If you add first, it is $7 \times 5$ and evaluates to 35.

If you multiply first, it is $3+20$ and evaluates to 23.

Is the first or second answer correct?

With no order of operations, both answers would be expected, but if an expression evaluates to more than one answer, math becomes ambiguous and does not work. For math to work there is only one order of operations to evaluate a mathematical expression.

The order of operations is Parenthesis, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This can be remembered in two ways: "Please Excuse My Dear Aunt Sally" or PEMDAS.

Please note that for Multiplication and Division and Addition and Subtraction you do which ever one comes first going from left to right and that is why both PEMDAS and BODMAS work. The following list from top to bottom is the order of operations in Algebra. Operations at the top of the list are completed first, operations on the same line are completed from left to right.

• Parenthesis ( )
• Exponent ^
• Multiply $\times$, Divide $\div$
• Add $+$, Subtract $-$

Parenthesis is a special operation that has the most precedence. You use the ( and ) signs to make a separate expression from a group of terms. You evaluate an expression in parenthesis first. You use parenthesis if you need to do an operation with less precedence first. If the term in Parenthesis is juxtaposed to a variable with no multiplication, then the variable and Parenthesis is ONE TERM. (example: $2x$ is ONE TERM so if $x=(1+2)$, then $2(1+2)$ is ONE TERM and takes precedence in the Order of Operations)

## Example ProblemsEdit

Let's evaluate these expressions.

 $5 + x^3$ where $x=2$ $=5 + 2^3$ $=5 + (2 \times 2 \times 2)$ $=5 + 8$ $=13$ $(5 + x)^3$ where $x=2$ $=(5 + 2)^3$ $=7^3$ $=343$ $\frac {6}{x} + 3$ where $x=3$ $=\frac {6}{3} + 3$ $=2 + 3$ $=5$ $\frac {6}{x+3}$ where $x=3$ $=\frac {6}{3+3}$ $=\frac {6}{6}$ $=1$ $8 - x + 2$ where $x = 3$ $8 - 3 + 2$ (evaluate the – operation first. – and + have the same precedence but is left-to-right) $5 + 2$ $7$ $8 - (x + 2)$ where $x = 3$ $8 - (3 + 2)$ $8 - 5$ $3$

Back to the first problem: Evaluate the expression $3 + 4 \times 5$.

There is only one answer, 23, because we multiply first.

If we want to add first, we can use parentheses.

If we write $(3 + 4) \times 5$, then we add first, and get 35.

## Practice ProblemsEdit

62 × (8 – 6) =

72

8 + 6 × 32 =

62

32 ÷ 23 + 4 =

8

8 + 32 ÷ 16 =

10

6 +(4 ÷ 2)2 * 8

38

 « Basic Algebra Order of Operations » Exponents and Powers Variables and Expressions