Rules for Order of Operations (PEMDAS) – Examples and Practice Problems

The order of operations are the rules that tell us which operations should be performed first when we have multiple operations in an expression. The order of operations tells us that we have to start with parentheses, then exponents, then multiplication and division, and finally addition and subtraction.

Here, we will learn about the order of operations in detail. We will explore the acronym PEMDAS, and we will apply it to solve some practice problems.

ALGEBRA
rules of order of operations

Relevant for…

Learning the correct order of operations to avoid incorrect answers.

See rules

ALGEBRA
rules of order of operations

Relevant for…

Learning the correct order of operations to avoid incorrect answers.

See rules

Mathematical operations

In mathematics, operations refer to things like adding, subtracting, multiplying, dividing, etc. When we add two numbers, we are performing the addition operation with those numbers. Similarly, when we multiply two numbers, we are performing the multiplication operation with those numbers.

Performing individual operations is very easy, but when we have something like the following expression, this can get a bit confusing:

$latex 2+\left( {{{4}^{2}}-2\times 3} \right)$

Which part should we calculate first?

Should we go from the left to the right?

Or should we go from the right to the left?

The answer to this is to use the order of operations.


Rules for order of operations

The order of operations are the rules that indicate the sequence in which multiple operations in an expression have to be solved.

1. Solve all the operations inside parentheses or other grouping signs:

order of operations example 1

2. Solve the exponents (powers, roots) before multiplying, dividing, adding or subtracting:

Order of operations 2

3. Multiply or divide before adding or subtracting:

Order of operations 3

4. The equivalent operations are solved from left to right:

Order of operations 4

Acronym PEMDAS

One way to easily remember the order of operations is PEMDAS, where each letter represents a mathematical operation:

P     Parenthesis

E     Exponents

M    Multiplication

D     Division

A     Addition

S     Subtraction

STEPS

1. We start with parentheses and other grouping signs.

2. We solve exponents, that is, roots and powers.

3. Since multiplication and division are on the same level, we solve from left to right.

4. Since additions and subtractions are at the same level, we solve from left to right.


Why follow the order of operations?

We must apply the correct order of mathematical operations when we solve expressions so that they always arrive at the same correct answer.

The following is an example of how we can get different answers if we don’t follow PEMDAS:

Expression solved from left to right

Order of operations 5

Expression solved with the order of operations

Order of operations 6

Examples with answers

Try to solve the exercises yourself and look at the process carefully to master the use of PEMDAS.

EXAMPLE 1

What is the result of $latex 5+4\times 3 -8$?

The order of operations tells us that we have to solve multiplication and division before addition and subtraction.

Also, when we have two or more operations of the same type, we solve from left to right. Thus, we have:

$latex 5+4\times 3 -8$

$latex =5+12 -8$

$latex =17-8$

$latex =9$

EXAMPLE 2

Find the result of $latex (4+8)-3\times 3-4$.

We solve the operation inside the parentheses first, then the multiplication, and we finish with the additions and subtractions:

$latex (4+8)-3\times 3-4$

$latex =(12)-3\times 3-4$

$latex =12-9-4$

$latex =-1$

EXAMPLE 3

Solve $latex 3+{{3}^{2}}\left( {3+4} \right)$.

We use PEMDAS to solve as follows:

$latex 3+{{3}^{2}}\left( {3+4} \right)$

$latex =3+{{3}^{2}}\left( 7 \right)$

$latex =3+9\left( 7 \right)$

$latex =3+63$

$latex =66$

EXAMPLE 4

Solve $latex \left( {3+4\times 3} \right)-{{2}^{2}}$.

By applying PEMDAS, we can solve as follows:

$latex \left( {3+4\times 3} \right)-{{2}^{2}}$

$latex =\left( {3+12} \right)-{{2}^{2}}$

$latex =15-{{2}^{2}}$

$latex =15-4$

$latex =11$

EXAMPLE 5

Solve $latex 10\times \left( {3+2} \right)+{{3}^{2}}\left( {3+4} \right)\div 3$.

We apply PEMDAS to solve:

$latex 10\times \left( {4+2} \right)+{{3}^{2}}\left( {3+4} \right)\div 3$

$latex =10\times 6+{{3}^{2}}\left( {3+4} \right)\div 3$

$latex =10\times 6+{{3}^{2}}\left( 7 \right)\div 3$

$latex =10\times 6+9\left( 7 \right)\div 3$

$latex =60+9\left( 7 \right)\div 3$

$latex =60+63\div 3$

$latex =60+21$

$latex =81$


Order of operations – Practice problems

Simplify the expression $latex 4+{{(-1(-2-1))}^2}$.

Choose an answer






Simplify the expression $latex 16-3{{(8-3)}^2}\div 5$.

Choose an answer






What is the result of $latex (10\times 5 -8)\div 6+3^2$?

Choose an answer







See also

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