Transcript Alg 1 - Ch 8.1 Multiplication Prop of Exponents

Algebra 1
Ch 8.1 – Multiplication Property of
Exponents
Objective
 Students will use the properties of exponents
to multiply exponential expressions
Before we begin
 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…
Review
4
5
Power or Exponent
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
Review – Common Error
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:
INCORRECT
5 ● 5 ● 5 ● 5 = 625
5 ● 4 = 20
One more thing…
 When working with exponents, the exponent
only applies to the number or variable directly
to the left of the exponent.
Example:
3x4y
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
Properties
 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…
Product of Powers Property
 To multiply powers having the same base, add
the exponents.
Example:
am ● an = am+n
Proof:
Three factors
a2 ● a3 = a ● a ● a ● a ● a = a2 + 3 = a5
Two factors
Example #1
53 ● 56
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:
53 ● 56 = 53+6 = 59
Example #2
x2 ● x3 ● 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:
x2 ● x3 ● x4 = x2+3+4 = x9
Power of a Power Property
 To find a power of a power, multiply the
exponents
Example:
(am)n = am●n
Proof:
Three factors
(a2)3 = a2●3 = a2 ● a2 ● a2 = a ● a ● a ● a ● a ● a = a6
Six factors
Example #3
(35)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:
(35)2 = 35●2 = 310
Example #4
[(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
Power of a Product Property
 To find a power of a product, find the power of
each factor and multiply
Example:
(a ● b)m = am ● bm
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
Example #5
(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 = 62 ● 52 = 36 ● 25 = 900
Example #6
(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 = 43y3z3 = 64y3z3
Example # 7
(-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 ● w2 = 4w2
Caution: It is expected that you know -22 = (-2)●(-2) = +4
Example #8
– (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 = – (22 ● w2) = – 4w2
Caution: In this example the negative sign is outside the brackets.
It does not mean that the 2 inside the parenthesis is negative!
Using all 3 properties
 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…
Example #9
Simplify
(4x2y)3 ● x5
I see that I have a power of a product in this expression
(4x2y)3
Let’s simplify that first by applying the exponent 3 to each term
within the parenthesis
(4x2y)3 ● x5 = 43 ●(x2)3 ● y3 ● x5
I now see that I have a power of a power in this expression
Let’s simplify that next by multiplying the exponents
= 43 ●(x2)3 ● y3 ● x5 = 43 ● x6 ● y3 ● x5
(x2)3
Example #9 (Continued)
= 4 3 ● x6 ● y3 ● x5
I now see that I have x6 and x5, so I will use the product of powers
property which states if the base is the same add the exponents.
Which looks like this:
= 43 ● x11 ● y3
All that’s left to do is simplify the term 43
= 64 ● x11 ● y3 = 64x11y3
Comments
 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 multi-step…
 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…
Comments
 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…
Your Turn
 Simplify the expressions
1. c ● c ● c
2. x4 ● x5
3. (43)3
4. (y4)5
5. (2m2)3
Your Turn
 Simplify the expressions
6. (x3y5)4
7. [(2x + 3)3]2
8. (3b)3 ● b
9. (abc2)3(a2b)2
10. –(r2st3)2(s4t)3
Your Turn Solutions
1. c3
6. x12y20
2. x9
7. (2x + 3)6
3. 49 or 262,144
8. 33B4 or 27b4
4. y20
9. a7b5c6
5. 8m6
10. -r4s14t9
Summary
 A key tool in making learning effective is being able to
summarize what you learned in a lesson in your own
words…
 In this lesson we talked about multiplication properties of
exponents. Therefore, in your own words summarize this
lesson…be sure to include key concepts that the lesson
covered as well as any points that are still not clear to
you…
 I will give you credit for doing this lesson…please see the
next slide…
Credit
 I will add 25 points as an assignment grade for you working on this
lesson…
 To receive the full 25 points you must do the following:



Have your name, date and period as well a lesson number as a
heading.
Do each of the your turn problems showing all work
Have a 1 paragraph summary of the lesson in your own words
 Please be advised – I will not give any credit for work submitted:



Without a complete heading
Without showing work for the your turn problems
Without a summary in your own words…