8.1: Exponent Properties Involving Products
Learning Objectives
 Use the product of a power property.
 Use the power of a product property.
 Simplify expressions involving product properties of exponents.
Introduction
In this chapter, we will discuss exponents and exponential functions. In Lessons 8.1, 8.2 and 8.3, we will be learning about the rules governing exponents. We will start with what the word exponent means.
Consider the area of the square shown right. We know that the area is given by:
But we also know that for any rectangle,
Similarly, the volume of the cube is given by:
But we also know that the volume of the cube is given by
You probably know that the power (the small number to the top right of the
Example 1
Write in exponential form.
(a)
(b)
(c)
(d)
Solution
(a)
(b)
(c)
(d)
When we deal with numbers, we usually just simplify. We'd rather deal with 16 than with
Let’s simplify Example 1 by evaluating the numbers.
Example 2
Simplify.
(a)
(b)
(c)
(d)
Solution
(a)
(b)
(c)
(d)
Note: You must be careful when taking powers of negative numbers. Remember these rules.
For even powers of negative numbers, the answer is always positive. Since we have an even number of factors, we make pairs of negative numbers and all the negatives cancel out.
For odd powers of negative numbers, the answer is always negative. Since we have an odd number of factors, we can make pairs of negative numbers to get positive numbers but there is always an unpaired negative factor, so the answer is negative:
Use the Product of Powers Property
What happens when we multiply one power of
So
So
You should see that when we take the product of two powers of
Product rule for exponents:
Example 3
Multiply
Solution
When multiplying exponents of the same base, it is a simple case of adding the exponents. It is important that when you use the product rule you avoid easytomake mistakes. Consider the following.
Example 4
Multiply
Solution
Note that when you use the product rule you DO NOT MULTIPLY BASES. In other words, you must avoid the common error of writing
Example 5
Multiply
Solution
In this case, the bases are different. The product rule for powers ONLY APPLIES TO TERMS THAT HAVE THE SAME BASE. Common mistakes with problems like this include
Use the Power of a Product Property
We will now look at what happens when we raise a whole expression to a power. Let’s take
So
When we take an expression and raise it to a power, we are multiplying the existing powers of
Power rule for exponents:
Power of a product
If we have a product inside the parenthesis and a power on the parenthesis, then the power goes on each element inside. So that, for example,
Power rule for exponents:
WATCH OUT! This does NOT work if you have a sum or difference inside the parenthesis. For example,
Let’s apply the rules we just learned to a few examples.
When we have numbers, we just evaluate and most of the time it is not really important to use the product rule and the power rule.
Example 6
Simplify the following expressions.
(a)
(b)
(c)
Solution
In each of the examples, we want to evaluate the numbers.
(a) Use the product rule first:
Then evaluate the result:
OR
We can evaluate each part separately and then multiply them.
(b) Use the product rule first.
Then evaluate the result.
OR
We can evaluate each part separately and then multiply them.
(c) Use the power rule first.
Then evaluate the result.
OR
We evaluate inside the parenthesis first.
Then apply the power outside the parenthesis.
When we have just one variable in the expression then we just apply the rules.
Example 7
Simplify the following expressions.
(a)
(b)
Solution
(a) Use the product rule.
(b) Use the power rule.
When we have a mix of numbers and variables, we apply the rules to the numbers or to each variable separately.
Example 8
Simplify the following expressions.
(a)
(b)
(c)
Solution
(a) We group like terms together.
We multiply the numbers and apply the product rule on each grouping.
(b) We groups like terms together.
We multiply the numbers and apply the product rule on each grouping.
(c) We apply the power rule for each separate term in the parenthesis.
We evaluate the numbers and apply the power rule for each term.
In problems that we need to apply the product and power rules together, we must keep in mind the order of operation. Exponent operations take precedence over multiplication.
Example 9
Simplify the following expressions.
(a)
(b)
(c)
Solution
(a)
We apply the power rule first on the first parenthesis.
Then apply the product rule to combine the two terms.
(b)
We must apply the power rule on the second parenthesis first.
Then we can apply the product rule to combine the two parentheses.
(c)
We apply the power rule on each of the parentheses separately.
Then we can apply the product rule to combine the two parentheses.
Review Questions
Write in exponential notation.

4⋅4⋅4⋅4⋅4 
3x⋅3x⋅3x 
(−2a)(−2a)(−2a)(−2a) 
6⋅6⋅6⋅x⋅x⋅y⋅y⋅y⋅y
Find each number:

54 
(−2)6 
(0.1)5 
(−0.6)3
Multiply and simplify.

63⋅66 
22⋅24⋅26 
32⋅43  \begin{align*}x^2 \cdot x^4\end{align*}
 \begin{align*}(2y^4) (3y)\end{align*}
 \begin{align*}(4a^2)(3a)(5a^4)\end{align*}
Simplify.
 \begin{align*}(a^3)^4\end{align*}
 \begin{align*}(xy)^2\end{align*}
 \begin{align*}(3a^2 b^3 )^4\end{align*}
 \begin{align*}(2xy^4 z^2)^5\end{align*}
 \begin{align*}(8x)^3(5x)^2\end{align*}
 \begin{align*}(4a^2)(2a^3)^4\end{align*}
 \begin{align*}(12xy)(12xy)^2\end{align*}
 \begin{align*}(2xy^2)(x^2 y)^2 (3x^2 y^2)\end{align*}
Review Answers
 \begin{align*}4^5\end{align*}
 \begin{align*}(3x)^3\end{align*}
 \begin{align*}(2a)^4\end{align*}
 \begin{align*}6^3 x^2 y^4\end{align*}
 625
 64
 0.00001
 0.216
 10077696
 4096
 576
 \begin{align*}x^6\end{align*}
 \begin{align*}6y^5 \end{align*}
 \begin{align*}60a^7\end{align*}
 \begin{align*}a^{12}\end{align*}
 \begin{align*}x^2 y^2 \end{align*}
 \begin{align*}81a^8 b^{12} \end{align*}
 \begin{align*}32x^5 y^{20} z^{10}\end{align*}
 \begin{align*}12800x^5\end{align*}
 \begin{align*}64a^{14}\end{align*}
 \begin{align*}1728x^3 y^3\end{align*}
 \begin{align*}6x^7 y^6\end{align*}
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