The order of operations

The order of operations tells us which calculations to perform first and which to perform afterward to reach the correct answer. It is essential both for solving mathematics problems and for building equations in mental arithmetic games. In this article, the Fastmath team explains the basics to help you solve future questions.

Parentheses, addition, subtraction, multiplication, and division

Let’s begin with the basic operations. Their order is:

1. Parentheses: Evaluate expressions inside parentheses first, working from the innermost pair outward.
2. Multiplication and division: Work from left to right.
3. Addition and subtraction: Work from left to right.

An example makes this clearer:

In the expression above, we calculate the multiplication and division first, then add and subtract the results.

Other operators

Mental arithmetic equations can also use operators such as:

• Exponentiation ($a^n$)
• Roots ($\sqrt{\ \ }$)
• Factorials ($!$)
• Summation ($\sum$)

Unless parentheses specify otherwise, evaluate these before addition, subtraction, multiplication, and division.

Let’s look at an example:

Notice that we evaluate these special operations first, followed by multiplication and division, then addition and subtraction.

When an equation contains parentheses, finish evaluating the expression inside each pair first. The same order of operations applies within them:

Knowing the order of operations helps you calculate correctly and build valid mental arithmetic equations. You may occasionally forget or mix up the order at first, but regular practice will help you remember and apply it confidently.