Transcript Ch. 8.1 Multiplication Prop of Exponents

Drill
Evaluate the expression.
1.
2.
3.
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 ● a2 + 3 = a5
a=
a
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 ● x4
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 ● a=2 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 ●
5
x
I now see that I have a power of a power in this expression (x2)3
Let’s simplify that next by multiplying the
exponents
= 43 ●(x2)3 ● y3 ● = 43 ●
x5
x6 ● y3 ● x5
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
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