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Order of Operations

Let’s review some basic expressions and their order of operations.

Understanding Order of Operations

In mathematics, the word "operations" means things like addition, subtraction, multiplication, division, squaring, etc. When evaluating an expression such as 8 - 3 x 5, in which order should one perform the operations? For example, should the subtraction be applied before the multiplication, or vice versa? In mathematics, we do the multiplication first, so the correct answer is -7. Sometimes brackets, which have their own rules, may be used to avoid confusion: 8 – (3 x 5). The standard order of operations, or precedence, is as follows.

  • Exponents and roots
  • Multiplication and division
  • Addition and subtraction

This means that if an expression is grouped by one or more symbols (one operator followed by another), the operator higher on the list should be applied first. Notice that multiplication and division are interchangeable since they are of equal precedence, as are addition and subtraction. Acronyms are often used to help students remember the rules (in this case, the order of operation); however, the acronym may vary depending on the instructor. The most common acronym for order of operations is BEDMAS, meaning

Brackets first, then Exponents, Division, Multiplication, Addition, and Subtraction. Some instructors may refer to Brackets as Parentheses, and Exponents may be called Indices, Powers, or Orders.

Example:

Evaluate the equation  \(3 \times 4 + 6 \div 2\).

Solution:

First we multiply the 3 by the 4:  \(3 \times 4 = 12\), which results in \(12 + 6 \div 2\).

Then we divide the 6 by the 2:  \(6 \div 2 = 3\), which results in \(3 + 12\).

Finally, we add the 12 and the 3 for a final answer of \(3 + 12 = 15\) 

 Let’s take a look at a more complicated example.

Example:

Evaluate the equation \(7+(6\times5^2+10)\div10\). 

Solution:

First we evaluate the exponent in the bracket: \(5^2 = 25\), which results in \(7 + (6 \times 25 + 10)\div10\).

Then we multiply the 6 by the 25 inside the bracket: \(6 \times 25 = 150\), which results in \(7 + (150 + 10) \div 10\).

Then we add the 10 to the 150 inside the bracket: \(150 + 10 = 160\), which results in \(7 + 160 \div10\).

Next, we divide the 160 by the 10: \(160 \div 10 = 16\), which results in \(7 + 16\).

Finally, we add the 16 to the 7 for a final answer of \(7 + 16 = 23\). 

Example:

Example 2: