Order of operations

The order of operations is the set of steps you use to simplify an expression.

The order of operations has four main steps:

1. Parentheses
2. Exponents
3. Multiplication and Division
4. Addition and Subtraction

Multiplication and division are listed together, and addition and subtraction are listed together. Within each pair, you work from left to right as the operations appear in the expression.

If you don’t follow this order, you can end up with an incorrect result.

Parentheses

Parentheses are the first thing to look for when you simplify an expression.

• If there are no parentheses, skip this step.
• If there are parentheses, evaluate the expression inside them first.
• If there are multiple sets of parentheses (for example, in a fraction), evaluate each set separately.

Let’s explore an example:

$$\frac{2\ast 4}{9-5}$$

You can think of the numerator and denominator as having their own parentheses:

$$\frac{(2\ast 4)}{(9-5)}$$

Now evaluate each set of parentheses:

$$\frac{(2\ast 4=8)}{(9-5=4)}=\frac{8}{4}$$

At this point, you’ve completed the parentheses step and simplified the expression to $8/4$. Now continue through the order of operations:

• There are no exponents, so skip that step.
• There’s no multiplication, but there is division.

Dividing $8$ by $4$ gives the final answer: $2$.

Another common situation is parentheses inside other parentheses. In that case, work from the inside out.

$$\begin{array}{c}2+4(4-2(3-2))\\ 2+4(4-2(1))\\ 2+4(4-2)\\ 2+4(2)\\ 2+8\\ 10\end{array}$$

Here, you start with the innermost parentheses, $(3-2)$, which equals $1$. Then you move outward step by step, applying the PEMDAS order within each set of parentheses.

Exponents

The exponent step means you simplify any exponents:

$$\begin{array}{c}3+4^2\\ 3+16\\ 19\end{array}$$

Because the exponent step comes before addition, you evaluate $4^2=16$ first, and then add to get $3+16=19$.

Going through the steps

No matter how easy or hard an expression is, always follow the order of operations (PEMDAS).

A good way to stay organized is to keep each step as simple as possible. Work one step at a time:

• Parentheses
• Exponents
• Multiplication and division (left to right)
• Addition and subtraction (left to right)

Parentheses don’t add new operations - they just tell you which part of the expression to simplify first.