Order of operations

The order of operations is the order of the steps you can 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, as are addition and subtraction. The order within these pairs doesn’t matter, i.e., you’ll still end up with the correct result if you do division first and then multiplication, or subtraction first and then addition.

If you don’t follow this order, you might end up with an incorrect result!

Parentheses

Parentheses are the first thing you look for when analyzing an expression. If there are no parentheses, then you can skip this step. If there are parentheses in the expression, be sure to evaluate the subexpression inside them first. If there are multiple sets of parentheses, like with a fraction, solve each individual set of parentheses separately. Let’s explore an example:

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

We can imagine there are parentheses around the numerator and denominator:

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

Now we can solve each set of parentheses individually:

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

We’ve completed the parentheses step and simplified this expression to $8/4$, so we can continue through the order of operations. There are no exponents, so we’ll skip that step. There is also no multiplication, but there is division. Dividing $8$ by $4$ results in our final answer: $2$.

Another common example of more difficult parentheses steps involves parentheses within other parentheses. In this situation, work from the inside out.

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

We started with the innermost parentheses, $(3-2)$, which equals $1$. Then we moved outward step by step, following the PEDMAS order within each parenthesis.

Exponents

The exponent step just means to simplify any exponents:

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

Since the exponent step comes before addition, we first evaluated $4^2=16$, and then added the result to get $3+16=19$.

Going through the steps

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

The most important thing when it comes to mastering PEMDAS is to keep everything simple. Don’t get discouraged by a difficult question, and don’t get overwhelmed by all the steps in the order of operations. Just go one step at a time: you know how to add, subtract, multiply, divide, and solve exponents. Adding parentheses just means you’ll need to work on one part of the equation before the rest.