The order of operations allows us to perform multiple algebraic operations in the correct order. Following the order of operations is extremely important, otherwise, we will end up with the wrong answer.
In this article, we will look at a summary of the order of operations along with examples with answers and exercises to solve.
Summary of order of operations
The order of operations are the rules that indicate the order in which we must solve the multiple operations in an algebraic expression. One way to remember the order of operations is with PEMDAS, where each letter represents a mathematical operation:
| P | Parentheses |
| E | Exponents |
| M | Multiplication |
| D | Division |
| A | Addition |
| S | Subtraction |
The order of operations tells us that the order in which we must solve the operations in an expression is:
1. Parentheses: Parentheses and other grouping signs take precedence over other operators.
2. Exponents: We solve all exponential and radical expressions, that is, powers and roots.
3. Multiplication and division: Multiplication and division are on the same level, so we solve from left to right when we have several multiplications or divisions.
4. Addition and subtraction: Addition and subtraction are on the same level, so we solve from left to right when we have several additions or subtractions.
10 Examples of order of operations with answers
EXAMPLE 1
Simplify the expression $latex 5+{{(3+1)}^2}$.
Solution
First, we have to simplify the expression that is inside the parentheses, then we apply the exponent and finally, we perform the addition:
$latex 5+{{(3+1)}^2}$
$latex =5+{{(4)}^2}$
$latex =5+16$
$latex =21$
EXAMPLE 2
Use the order of operations to solve $latex 5\times 4^2 -8\times 2+5$.
Solution
The order of operations tells us that we have to solve the exponents first, then the multiplications from left to right.
Finally, we solve the additions and subtractions also from left to right:
$latex 5\times 4^2 -8\times 2+5$
$latex =5\times 16 -8\times 2+5$
$latex =80 -8\times 2+5$
$latex =80-16+5$
$latex =64+5$
$latex =69$
EXAMPLE 3
Find the result of $latex 5(2^2-5)+4\times 3^2-15\times 2$.
Solution
Let’s start by solving the operations inside the parentheses. Then we solve exponents, multiplications, and addition and subtraction, in that order.
$latex 5(2^2-5)+4\times 3^2-15\times 2$
$latex =5(4-5)+4\times 3^2-15\times 2$
$latex =5(-1)+4\times 3^2-15\times 2$
$latex =5(-1)+4\times 9-15\times 2$
$latex =-5+4\times 9-15\times 2$
$latex =-5+36-15\times 2$
$latex =-5+36-30$
$latex =31-30$
$latex =1$
EXAMPLE 4
Simplify the expression $latex 5+{{[-2(-1+3)]}^2}$.
Solution
When we have multiple grouping signs, we start with the inner parentheses and solve outward.
First, we solve the expression inside the parentheses, then, we solve the expression inside the brackets. After that, we apply the exponent and finish with the addition:
$latex 5+{{[-2(-1+3)]}^2}$
$latex =5+{{[-2(2)]}^2}$
$latex =5+{{[-4]}^2}$
$latex =5+16$
$latex=21$
EXAMPLE 5
Solve the expression $latex 4-2[5-2(4-2)]\div 2$.
Solution
We simplify from the inside out, first the parentheses and then the brackets. After simplifying the grouping signs, we perform the division, followed by the addition of 4:
$latex 4-2[5-2(4-2)]\div 2$
$latex =4-2[5-2(2)]\div 2$
$latex =4-2[5-4]\div 2$
$latex =4-2[1]\div 2$
$latex =4-2\div 2$
$latex =4-1$
$latex =3$
EXAMPLE 6
What is the result of $latex 12-2{{(6-3)}^2}\div 3$?
Solution
We start by solving the expression inside the parentheses and then applying the exponent. Then, we do the multiplication by -2, followed by division by 3 to finish adding 12:
$latex 12-2{{(6-3)}^2}\div 3$
$latex =12-2{{(3)}^2}\div 3$
$latex =12-2(9)\div 3$
$latex =12-18\div 3$
$latex =12-6$
$latex =6$
EXAMPLE 7
Simplify the expression $latex 12x+4[6-(3x+2)]$.
Solution
In this case, we have the variable x. The order of operations is the same, with the only difference that we must combine only like terms. Therefore, we start by expanding the parentheses. Then, we simplify the expression inside the brackets and then multiply this expression by 4. We finish by adding 12 x:
$latex 12x+4[6-(3x+2)]$
$latex =12x+4[6-3x-2)]$
$latex 12x+4[-3x+4]$
$latex =12x-12x+16$
$latex =16$
EXAMPLE 8
Find the result of the expression $latex -\{4x-[4-(3-2x)]+5x\}$.
Solution
We start by expanding the parentheses by applying the negative sign. Then, we combine like terms in the expression inside the brackets and apply the negative sign. Then, we combine like terms in the expression inside the curly braces and end up applying the negative sign:
$latex -\{4x-[4-(3-2x)]+5x\}$
$latex =-\{4x-[4-3+2x)]+5x\}$
$latex =-\{4x-[1+2x]+5x\}$
$latex =-\{4x-1-2x+5x\}$
$latex =-\{7x-1\}$
$latex =-7x+1$
EXAMPLE 9
Apply the order of operations to solve the following
$$\frac{55}{6(3-2)+5}+\frac{2(3)^2}{8-2}$$
Solution
We must start by applying the order of operations individually both in the numerator and in the denominator of each fraction. Then, we simplify the fractions and end by adding the resulting expression:
$$\frac{55}{6(3-2)+5}+\frac{2(3)^2}{8-2}$$
$$=\frac{55}{6(1)+5}+\frac{2(9)}{6}$$
$$=\frac{55}{6+5}+\frac{18}{6}$$
$$=\frac{55}{11}+3$$
$latex =5+3$
$latex =8$
EXAMPLE 10
Simplify the expression using the order of operations
$$\frac{(5-4)+(4-1)^2}{5+(4-1)}$$
Solution
This example is similar to the previous one since we must apply the order of operations individually to the numerator and the denominator to then simplify the fraction:
$$\frac{(5-4)+(4-1)^2}{8+(2-5)}$$
$$=\frac{(1)+(3)^2}{8+(-3)}$$
$$=\frac{1+9}{5}$$
$$=\frac{10}{5}$$
$latex =2$
Order of operations – Practice problems
What is the result of the following? $$3(3^2+3)-14+4(5-2)^2-21\times 5$$
Write the answer in the input box.
See also
Interested in learning more about algebraic expressions? Take a look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
