Ch 8.1 – Multiplication Property of
Exponents

 Students will use the properties of exponents to multiply exponential expressions

 In chapter 8 we will be looking at exponents and exponential functions…
 That is, we will be looking at how to add, subtract, multiply and divide exponents…
 Once we have done that…we will apply what we have learned to simplifying expressions and solving equations…
 Before we do that…let’s do a quick review of what exponents are and how they work…

Power or Exponent
5 4
Base
The above number is an exponential expression.
The components of an exponential expression contain a base and a power
The power (exponent) tells the base how many times to multiply itself
In this example the exponent (4) tells the base (5) to multiply itself 4 times and looks like this:
5 ● 5 ● 5 ● 5

5 4
A common error that student’s make is they multiply the base times the exponent. THAT IS INCORRECT!
Let’s make a comparison:
Correct: 5 ● 5 ● 5 ● 5 = 625
INCORRECT 5 ● 4 = 20

 When working with exponents, the exponent only applies to the number or variable directly to the left of the exponent.
Example: 3x 4 y
In this example the exponent (4) only applies to the x
 If you have an expression in brackets. The exponent applies to each term within the brackets
Example: (3x) 2
In this example the exponent (2) applies to the 3 and the x

 In this lesson we will focus on the multiplication properties of exponents…
 There are a total of 3 properties that you will be expected to know how to work with. They are:
 Product of Powers Property
 Power of a Power Property
 Power of a Product Property


This gets confusing for students because all the names sound the same…
Let’s look at each one individually…

 To multiply powers having the same base, add the exponents.
Example: a m ● a n = a m+n
Proof:
Three factors a 2 ● a 3 = a ● a ● a ● a ● a = a 2 + 3 = a 5
Two factors

5 3 ● 5 6
When analyzing this expression, I notice that the base (5) is the same.
That means I will use the Product of Powers Property , which states when multiplying, if the base is the same add the exponents.
Solution:
5 3 ● 5 6 = 5 3+6 = 5 9

x 2 ● x 3 ● x 4
When analyzing this expression, I notice that the base (x) is the same.
That means I will use the Product of Powers Property , which states when multiplying, if the base is the same add the exponents.
Solution: x 2 ● x 3 ● x 4 = x 2+3+4 = x 9

 To find a power of a power, multiply the exponents
Example:
(a m ) n = a m ●n
Proof:
Three factors
(a 2 ) 3 = a 2
●3
= a 2 ● a 2 ● a 2 = a ● a ● a ● a ● a ● a = a 6
Six factors

(3 5 ) 2
When I analyze this expression, I see that I am multiplying exponents
Therefore, I will use the Power of a Power Property to simplify the expression, which states to find the power of a power, multiply the exponents.
Solution:
(3 5 ) 2 = 3 5 ●2
= 3 10

[(a + 1) 2 ] 5
When I analyze this expression, I see that I am multiplying exponents
Therefore, I will use the Power of a Power Property to simplify the expression, which states to find the power of a power, multiply the exponents.
Solution:
[(a + 1) 2 ] 5 = (a + 1) 2 ●5
= (a + 1) 10

 To find a power of a product, find the power of each factor and multiply
Example:
(a ● b) m = a m ● b m
This property is similar to the distributive property that you are expected to know. In this property essentially you are distributing the exponent to each term within the parenthesis

(6 ● 5) 2
When I analyze this expression, I see that I need to find the power of a product
Therefore, I will use the Power of a Product Property , which states to find the power of a product, find the power of each factor and multiply
Solution:
(6 ● 5) 2 = 6 2 ● 5 2 = 36 ● 25 = 900

(4yz) 3
When I analyze this expression, I see that I need to find the power of a product
Therefore, I will use the Power of a Product Property , which states to find the power of a product, find the power of each factor and multiply
Solution:
(4yz) 3 = 4 3 y 3 z 3 = 64y 3 z 3

(-2w) 2
When I analyze this expression, I see that I need to find the power of a product
Therefore, I will use the Power of a Product Property , which states to find the power of a product, find the power of each factor and multiply
Solution:
(-2w) 2 = (-2 ● w) 2 = (-2) 2 ● w 2 = 4w 2
Caution: It is expected that you know -2 2 = (-2) ●(-2) = +4

– (2w) 2
When I analyze this expression, I see that I need to find the power of a product
Therefore, I will use the Power of a Product Property , which states to find the power of a product, find the power of each factor and multiply
Solution:
– (2w) 2 = – (2 ● w) 2 = – (2 2 ● w 2 ) = – 4w 2
Caution: In this example the negative sign is outside the brackets.
It does not mean that the 2 inside the parenthesis is negative!

 Ok…now that we have looked at each property individually…
 let’s apply what we have learned and look at simplifying an expression that contains all 3 properties
 Again, the key here is to analyze the expression first…

Simplify (4x 2 y) 3 ● x 5
I see that I have a power of a product in this expression
(4x 2 y) 3
Let’s simplify that first by applying the exponent 3 to each term within the parenthesis
(4x 2 y) 3 ● x 5 = 4 3 ●(x 2 ) 3 ● y 3 ● x 5
I now see that I have a power of a power in this expression
(x 2 ) 3
Let’s simplify that next by multiplying the exponents
= 4 3 ●(x 2 ) 3 ● y 3 ● x 5 = 4 3 ● x 6 ● y 3 ● x 5

= 4 3 ● x 6 ● y 3 ● x 5
I now see that I have x 6 and x 5 , so I will use the product of powers property which states if the base is the same add the exponents.
Which looks like this:
= 4 3 ● x 11 ● y 3
All that’s left to do is simplify the term 4 3
= 64 ● x 11 ● y 3 = 64x 11 y 3

 These concepts are relatively simple…
 As you can see, to be successful here the key is to analyze the expression first…and then lay out your work in an organized step by step fashion…as I have illustrated.
 As a reminder, for the remainder of this course all the problems will be multistep…
 Therefore, you will be expected to know these properties and apply them in different situations later on in the course when we work with polynomials and factoring…

 On the next couple of slides are some practice problems…The answers are on the last slide…
 Do the practice and then check your answers…If you do not get the same answer you must question what you did…go back and problem solve to find the error…
 If you cannot find the error bring your work to me and I will help…

 Simplify the expressions
1.
2.
3.
4.
5.
c ● c ● c x 4 ● x 5
(4 3 ) 3
(y 4 ) 5
(2m 2 ) 3

 Simplify the expressions
6.
7.
8.
(x 3 y 5 ) 4
[(2x + 3) 3 ] 2
(3b) 3 ● b
9.
10.
(abc 2 ) 3 (a 2 b) 2
–(r 2 st 3 ) 2 (s 4 t) 3