8.1: Exponential Properties Involving Products
In this lesson, you will be learning what an exponent is and about the properties and rules of exponents. You will also learn how to use exponents in problem solving.
Definition: An exponent is a power of a number that shows how many times that number is multiplied by itself.
An example would be
Example 1: Write in exponential form:
Solution: You must count the number of times the base,
Note: There are specific rules you must remember when taking powers of negative numbers.
(negative \ number) \times (negative \ number) &= positive \ number
For even powers of negative numbers, the answer will always be positive. Pairs can be made with each number and the negatives will be cancelled out.
For odd powers of negative numbers, the answer is always negative. Paris can be made but there will still be one negative number unpaired, making the answer negative.
When we multiply the same numbers, each with different powers, it is easier to combine them before solving. This is why we use the Product of Powers Property.
Product of Powers Property: For all real numbers
Example 2: Multiply
Solution:
Note that when you use the product rule you DO NOT MULTIPLY BASES.
Example:
Another note is that this rule APPLIES ONLY TO TERMS THAT HAVE THE SAME BASE.
Example:
& \underbrace{(x \cdot x \cdot x \cdot x}_{x^4}) \cdot \underbrace{(x \cdot x \cdot x \cdot x}_{x^4}) \cdot \underbrace{(x \cdot x \cdot x \cdot x}_{x^4})=\underbrace{(x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x)}_{x^{12}}
This situation is summarized below.
Power of a Product Property: For all real numbers
The Power of a Product Property is similar to the Distributive Property. Everything inside the parentheses must be taken to the power outside. For example,
The Power of a Product Property does not work if you have a sum or difference inside the parenthesis. For example,
Example 3: Simplify
Solution:
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.
CK12 Basic Algebra: Exponent Properties Involving Products (14:00)
 Consider
a5 . 1. What is the base? 2. What is the exponent? 3. What is the power? 4. How can this power be written using repeated multiplication?
Determine whether the answer will be positive or negative. You do not have to provide the answer.

−(34) 
−82 
10×(−4)3  What is the difference between
−52 and(−5)2 ?
Write in exponential notation.

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

110 
03 
73 
−62 
54 
34⋅37 
26⋅2 
(42)3 
(−2)6 
(0.1)5 
(−0.6)3
Multiply and simplify.

63⋅66 
22⋅24⋅26 
32⋅43 
x2⋅x4 
x2⋅x7 
(y3)5 
(−2y4)(−3y) 
(4a2)(−3a)(−5a4)
Simplify.

(a3)4 
(xy)2 
(3a2b3)4 
(−2xy4z2)5 
(3x2y3)⋅(4xy2) 
(4xyz)⋅(x2y3)⋅(2yz4) 
(2a3b3)2 
(−8x)3(5x)2 
(4a2)(−2a3)4 
(12xy)(12xy)2 
(2xy2)(−x2y)2(3x2y2)
Mixed Review
 How many ways can you choose a 4person committee from seven people?
 Three canoes cross a finish line to earn medals. Is this an example of a permutation or a combination? How many ways are possible?
 Find the slope between (–9, 11) and (18, 6).
 Name the number set(s) to which
36−−√ belongs.  Simplify
74x2−−−−√ .  78 is 10% of what number?
 Write the equation for the line containing (5, 3) and (3, 1).