This page of our ParaPro study guide covers the order of operations. We teach you the step-by-step method to apply the order of operations when solving equations. However, you need to understand exponents as part of this process, so we will start there.
Exponents
An exponent represents repeated multiplication. For example, $2^5$ means $2*2*2*2*2=32$.
In this example, $2$ is known as the base. The $5$, written at the top right of the base, is known as the exponent.
$base^{exponent}$
$2^5$ is read out loud as “$2$ raised to the $5^{th}$ power,” “$2$ to the power of $5$,” or just “$2$ to the $5^{th}$.”
When a number is squared, it is raised to the power of $2$. Mathematically, if $n$ is any number, then $n$ squared is written as $n^²$.
- This means $n^2 = n \times n$.
- For example, $3^2 = 3 \times 3 = 9$.
When a number is cubed, it is raised to the power of $3$. Mathematically, if $n$ is any number, then $n$ cubed is written as $n^3$.
- This means $n^3 = n \times n \times n$.
- For example, $2^3 = 2 \times 2 \times 2 = 8$.
A number raised to the power of $1$ equals itself. For example:
$6^1 = 6$
A number raised to the power of $0$ equals $1$. For example:
$6^0 = 1$
When multiplying two powers with the same base, you can add the exponents. For example:
$6^3 \times 6^4$
$= 6^{3 + 4}$
$= 6^7$
Example 1
Evaluate $5^3$.
$5^3 = 5 \times 5 \times 5 = 125$
Example 2
Insert the correct symbol to compare the two values.
$3^2$_______$2^3$
- $<$
- $=$
- $>$
Solution: $3^2$ means $3*3=9$
$2^3$ means $2*2*2=8$
$3^2 > 2^3$
Example 3
Evaluate $2^3 \times 2^2$.
Solution: Using the multiplication rule: $2^3 \times 2^2 = 2^{3+2} = 2^5 = 32$
Order of Operations
When you have an expression with multiple operations, you have to apply them in the following order:
- Grouping symbols (parentheses, brackets, etc.)
- Exponents
- Multiplication and division (from left to right)
- Addition and subtraction (from left to right)
This is often referred to as PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).
Example 4
$4(2+9)÷4+3$
Step #1: Grouping symbols
$4(11)÷4+3$
Step #2: Exponents
There are no exponents in this expression.
Step #3: Multiplication and division (from left to right)
Remember that a number directly outside parentheses is to be multiplied by what is in the parentheses. Thus, we first multiply $4 * 11$.
$44÷4+3$
$11+3$
Step #4: Addition and subtraction (from left to right)
$14$
Example 5
$2^3$ ∙ $(1+3)-(2$ ∙ $5)$
Step #1: Grouping symbols
$2^3$ ∙ $(4)-(10)$
Step #2: Exponents
$8$ ∙ $(4)-(10)$
Step #3: Multiplication and division (from left to right)
$32-(10)$
Step #4: Addition and subtraction (from left to right)
$22$
Example 6
$8-6(4÷4)+3^2$
Step #1: Grouping symbols
$8-6(1)+3^2$
Step #2: Exponents
$8-6(1)+9$
Step #3: Multiplication and division (from left to right)
$8-6+9$
Step #4: Addition and subtraction (from left to right)
$2+9$
$11$
Example 7
$2^6-(4$ ∙ $5)(2)$
Step #1: Grouping symbols
$2^6-(20)(2)$
Step #2: Exponents
$64-(20)(2)$
Step #3: Multiplication and division (from left to right)
Remember that two parentheses next to each other will be multiplied by each other.
$64-40$
Step #4: Addition and subtraction (from left to right)
$24$
You should now understand that exponents represent repeated multiplication and know how to evaluate them. You should also understand the order of operations and remember what PEMDAS stands for. Before you move on to the next page, be sure to try the review test below.