1.7: Use the Order of Operations to Evaluate Powers
Do you know how to evaluate a variable expression when it includes powers? Take a look at this problem.
Evaluating dilemmas like this one is just what this Concept is all about. Pay attention and you will be able to work through this problem at the end of the Concept.
Guidance
Did you know that you can apply the order of operations to expressions that have powers in them?
Let's look at how to do this.
To do this, we are going to need to refer back to the order of operations.
Order of Operations
P parentheses or grouping symbols
E exponents
MD multiplication and division in order from left to right
AS addition and subtraction in order from left to right
Now look at the E. That E refers to exponents and powers and evaluating exponents in the order of operations. You can see that you evaluate the powers right after the grouping symbols.
It is a bit like working on a puzzle. Here is an expression that needs evaluating.
Evaluate the expression
It does look complicated, but if it helps, think of this as a series of steps. The order of operations is your guide. If you follow the order of operations then working through a problem such as this one becomes much easier.
The answer is 126.
We can also evaluate variable expressions that have more than one variable. Notice that a different value has been given for
Evaluate the expression
The answer is 117.
When you have variable and numerical expressions with powers in them, you can use the order of operations to evaluate the expressions. Remember not to get stuck if the problem seems complicated. Stick to the order of operations and you will be able to evaluate the expression.
Example A
Evaluate the expression
Solution:
Example B
Evaluate the expression
Solution:
Example C
Evaluate the expression
Solution:
Now let's go back to the dilemma from the beginning of the Concept. Evaluate this expression.
First, let's substitute the given values into the expression for
Now we can evaluate the powers. Here is our answer so far.
This is our final answer.
Vocabulary
 Numerical Expression
 a group of numbers and operations used to represent a quantity without an equals sign.
 Variable Expression
 a group of numbers, operations and variables used to represent a quantity without an equals sign.
 Powers
 the value of a base and an exponent.
 Base
 the regular sized number that the exponent works upon.
 Exponent
 the little number that tells you how many times to multiply the base by itself.
Guided Practice
Here is one for you to try on your own.
Evaluate the expression
Solution
Step 1: Before performing the order of operations, substitute 6 for “
Step 2: Perform the calculations inside the parentheses.
Step 3: Perform the calculations with exponents.
Step 4: Multiply
Step 5: Add
The answer is 1,464.
Video Review
Khan Academy Introduction to the Order of Operations
Practice
Directions: Evaluate each expression. Remember to follow the order of operations.

32+[(5⋅2)−3]−8⋅2 
52+(3+5)−62+2 
63+52+25 
16(123) 
82−(2(33)÷2)+(16⋅5)
Directions: Evaluate each expression by substituting the given value into each expression. Remember to follow the order of operations.
6.
7.
8.
9.
10.
11. \begin{align*}45 + 8y + 3^3\end{align*} for \begin{align*}y=5\end{align*}.
12. \begin{align*}3^3 + 8x 2^2\end{align*} for \begin{align*}x=7\end{align*}.
13. \begin{align*}(12)^2 + 7y 4^2\end{align*} for \begin{align*}y=6\end{align*}.
14. \begin{align*}4^3 + 9x + 11\end{align*} for \begin{align*}x=4\end{align*}.
15. \begin{align*}(7)^2 + 7x^2 + 12^2\end{align*} for \begin{align*}y=2\end{align*}.
16. \begin{align*}45 + 7^2  x^3\end{align*} for \begin{align*}x=4\end{align*}.
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Term  Definition 

Base  When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression , 32 is the base, and 4 is the exponent. 
Evaluate  To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value. 
Exponent  Exponents are used to describe the number of times that a term is multiplied by itself. 
Numerical expression  A numerical expression is a group of numbers and operations used to represent a quantity. 
Variable Expression  A variable expression is a mathematical phrase that contains at least one variable or unknown quantity. 
Image Attributions
Here you'll use the order of operations to evaluate powers.