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

## Vocabulary

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
• Subtraction

In the United States, you may also see PEMDAS:

• Parentheses
• Exponents
• Multiplication
• Division
• Subtraction

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

## Lesson

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 whichever 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, and 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 you treat this implicit multiplication the same as any other multiplication. (example: In $1/2x$ , the $2x$ is a juxtaposed, implicit multiplication, so it means the same as $1/2\times x$ , and multiplication does not take precedence over the division in the Order of Operations. If you want $2x$ to be a proper term, you should write $1/(2x)$ )

## Example problems

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 problems

Evaluate the following expressions:

1

 62 × (8 – 6) =

2

 8 + 6 × 32 =

3

 32 / 23 + 4 =

4

 8 + 32 / 16 =

5

 6 +(4 / 2)2 × 8=

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