Order of Operations

The order of operations might sound intimidating to a math learner who is encountering this collection of rules for the first time. But understanding the order of operations can make solving math expressions that involve multiple operations much easier. If you encounter an expression like the following, you might be uncertain of how to start:

$5+3\times {\left(8–9\right)}^2$

Fortunately, you can rely on PEMDAS to help you determine the correct order of operations when working through this type of math expression.

Solve expressions correctly using PEMDAS

PEMDAS stands for Parentheses, Exponents, Multiplication-Division, and Addition-Subtraction. You'll want to remember this acronym (in this order) when determining how to solve math expressions involving several operations (if you have a hard time remembering PEMDAS, you might try Please Excuse My Dear Aunt Sally!). If you have an expression like the one above, relying on PEMDAS will help you solve it correctly. Here's a breakdown of the process:

Start by making calculations inside the parentheses

When solving an expression with multiple operations, you'll want to first complete the calculations inside parentheses or other grouping symbols (ex. fraction bars -, brackets [], or braces {}) then work your way out. For example, for the problem $5+3\times {\left(8–9\right)}^2$ , first, calculate within the parentheses:

$5+3\times {\left(-1\right)}^2$

Simplify exponential expressions

Next, you'll want to simplify any exponents. The one exponent you'll find in this expression is ${\left(-1\right)}^2$ or $\left(-1\right)\times \left(-1\right)$ , which equals 1. Here's our updated problem:

$5+3\times 1$

Work multiplication and division

If your expression has multiplication or division components, you'll work on those from left to right. In this case, you'll multiply $3\times 1$ , which equals 3. Now, you're left with the following:

$5+3$

Finish by working addition and subtraction

Your final step will be to work on the addition or subtraction components of the expression:

$5+3=8$

You're all done!

$5+3\times {\left(8–9\right)}^2$

Pay close attention when solving expressions using PEMDAS

When working through expressions that have multiple operations, it's important to look closely at the structure of the problem before solving it. You could have two problems that look very similar but have different outcomes:

$7+8\times {\left(1+2\right)}^2=79$ vs. $7+8\times 1+2^2=19$

For the first problem, $7+8\times {\left(1+2\right)}^2$ , you: 1) make calculations within the parentheses, 2) simplify the exponential expression, 3) work the multiplication/division portion of the problem, and 4) work the addition/subtraction portion of the problem.

For the second problem, $7+8\times 1+2^2$ , you have no parentheses, so you

1) simplify the exponential expression.

2) work the multiplication/division portion of the problem.

3) work the addition/subtraction portion of the problem.

Using the order of operations with fractions

When it comes to fractions, you'll want to make sure to first simplify the numerator and denominator since the fraction bar serves as a grouping symbol. Then you can complete the division portion of the problem:

$3+\frac{5}{6}-2=\frac{8}{4}=2$

Practice questions using the order of operations

a. Work through an expression with parentheses, multiplication, and addition:

$\left(3\times 2\right)+5\left(2+4\right)$

$\left(3\times 2\right)+5\left(2+4\right)=36$

b. Work through an expression with exponents, multiplication, and subtraction:

$4^2-5\times 3^2$

$4^2-5\times 3^2=-29$

c. Work through an expression with parentheses, exponents, and multiplication:

${\left(2\times 2\right)}^2\times 5^2$

${\left(2\times 2\right)}^2\times 5^2=400$

d. Work through an expression with parentheses, exponents, multiplication, addition, and subtraction:

${\left(9-6\right)}^2+{\left(2\times 3\right)}^2$

${\left(9-6\right)}^2+{\left(2\times 3\right)}^2=45$

e. Work through an expression with division, addition, and subtraction:

$\frac{\left(9+6\right)}{\left(5-2\right)}$

$\frac{15}{3}=5$

f. Work through an expression with parentheses, multiplication, division, addition, and subtraction:

$2\left(1+3\right)+\frac{4}{7-3}$

$2\left(4\right)+\frac{4}{4}$

$8+1=9$

Topics related to the Order of Operations

Commutative Property

Distributive Property

Associative Property

Flashcards covering the Order of Operations

5th Grade Math Flashcards

Common Core: 5th Grade Math Flashcards

Practice tests covering the Order of Operations

MAP 5th Grade Math Practice Tests

Common Core: 5th Grade Math Diagnostic Tests

Get help learning how to solve expressions using the order of operations

While working through expressions using PEMDAS is a mostly straightforward process, it's easy to see how a student could get vastly different answers by making calculations out of order. If your student is having a hard time grasping the order of operations or would like additional practice with PEMDAS problems, working with a tutor could make a positive difference. Get more information about how tutoring can help your student with math by connecting with the Educational Directors at Varsity Tutors.