Transcript Simplifying Exponents

Simplifying
Exponents
Algebra I
1
Contents
•
•
•
•
Multiplication Properties of Exponents ……….1 – 13
Zero Exponent and Negative Exponents……14 – 24
Division Properties of Exponents ……………….15 – 32
Simplifying Expressions using Multiplication and
Division Properties of Exponents…………………33 – 51
• Scientific Notation ………………………………………..52 - 61
2
Multiplication Properties
of Exponents
• Product of Powers Property
• Power of a Power Property
• Power of a Product Property
3
Product of Powers
Property
• To multiply powers
that have the same
base, you add the
exponents.
• Example:
a a  aaaaa  a
2
3
23
a
5
4
Practice Product of
Powers Property:
•
•
Try:
x x
Try:
n n n
5
5
4
2
3
5
Answers To Practice
Problems
1.
Answer:
2. Answer:
x x  x
5
4
5 4
x
9
n 5  n 2  n 3  n 5 2  3  n10
6
Power of a Power
Property
• To find a power of a power, you
multiply the exponents.
• Example:
• Therefore,
(a )  a  a  a  a
2 3
2
2
2
2 2 2
a
6
(a 2 )3  a 23  a 6
7
Practice Using the Power
of a Power Property
1.
Try:
2. Try:
4 4
(p )
4 5
(n )
8
Answers to Practice
Problems
1.
Answer:
2. Answer:
( p 4 ) 4  p 44  p16
(n )  n
4 5
45
n
20
9
Power of a Product
Property
• To find a power of a product, find
the power of EACH factor and
multiply.
• Example: (4 yz )  4  y  z  64 y z
3
3
3
3
3 3
10
Practice Power of a
Product Property
1.
Try:
2. Try:
(2mn)
6
(abc)
4
11
Answers to Practice
Problems
1.
Answer:
2. Answer:
(2mn)  2 m n  64m n
6
6
6
6
6
6
(abc)  a b c
4
4 4 4
12
Review Multiplication
Properties of Exponents
• Product of Powers Property—To multiply powers
that have the same base, ADD the exponents.
• Power of a Power Property—To find a power of a
power, multiply the exponents.
• Power of a Product Property—To find a power of a
product, find the power of each factor and
multiply.
13
Zero Exponents
• Any number, besides zero, to the
zero power is 1.
• Example:
a 1
• Example:
4 1
0
0
14
Negative Exponents
• To make a negative
exponent a positive
exponent, write it as
its reciprocal.
• In other words, when
faced with a negative
exponent—make it
happy by “flipping” it.
15
Negative Exponent
Examples
• Example of Negative
Exponent in the
Numerator:
• The negative exponent
is in the numerator—
to make it positive, I
“flipped” it to the
denominator.
x
3
1
 3
x
16
Negative Exponents
Example
• Negative Exponent in
the Denominator:
• The negative exponent
is in the denominator,
so I “flipped” it to the
numerator to make
the exponent positive.
4
1
y
4


y
4
y
1
17
Practice Making Negative
Exponents Positive
1.
Try:
2. Try:
d
3
1
5
z
18
Answers to Negative
Exponents Practice
1.
Answer:
2. Answer:
d
3
1
 3
d
5
1
z
5

z
5
z
1
19
Rewrite the Expression
with Positive Exponents
• Example:
2x
3
y
2
• Look at EACH factor and decide if the factor belongs in the
numerator or denominator.
• All three factors are in the numerator. The 2 has a positive
exponent, so it remains in the numerator, the x has a
negative exponent, so we “flip” it to the denominator. The y
has a negative exponent, so we “flip” it to the denominator.
3
2x y
2
2

xy
20
Rewrite the Expression
with Positive Exponents

3
3

8
• Example:
4 ab c
•
All the factors are in the numerator.
Now look at each factor and decide if the
exponent is positive or negative. If the
exponent is negative, we will flip the
factor to make the exponent positive.
21
Rewriting the Expression
with Positive Exponents
3
3 8
•
Example:
•
The 4 has a negative exponent so to make the exponent positive—
flip it to the denominator.
•
The exponent of a is 1, and the exponent of b is 3—both positive
exponents, so they will remain in the numerator.
•
The exponent of c is negative so we will flip c from the numerator
to the denominator to make the exponent positive.
4 ab c
3
3
ab
ab

3 8
8
4 c
64c
22
Practice Rewriting the
Expressions with Positive
Exponents:
1.
Try:
2. Try:
1
2
3
3 x y z
2 3 4
4a b c d
23
Answers
1.
Answer
2. Answer
3
1
x
2
y
3
z 
z
3x 2 y 3
3
4
b
d
2 3 4
4a b c d  2 4
a c
24
Division Properties of
Exponents
• Quotient of Powers Property
• Power of a Quotient Property
25
Quotient of Powers
Property
• To divide powers
that have the same
base, subtract the
exponents.
• Example:
5
5 3
x
x
2


x
3
x
1
26
Practice Quotient of
Powers Property
1.
Try:
9
a
a3
3
2. Try:
y
4
y
27
Answers
1.
Answer:
a9
a 9 3
6


a
a3
1
3
2. Answer:
y
1
1
 4 3 
4
y
y
y
28
Power of a Quotient
Property
• To find a power of a
quotient, find the
power of the
numerator and the
power of the
denominator and
divide.
• Example:
a


b 
3
3
a

b3
29
Simplifying Expressions
 2m n

 3mn
3
• Simplify
4



3
30
Simplifying Expressions
• First use the Power of a Quotient
Property along with the Power of a Power
Property
 2m n

 3mn
3
4
3

2 m n
2 m n
  3 3 3  3 3 3
3 m n
3 m n

3
33
43
3
9
12
31
Simplify Expressions
• Now use the
Quotient of Power
Property
3
9 12
9 3 123
2 mn
8m n

3
3 3
3mn
27
6
8m n

27
9
32
Simplify Expressions
• Simplify
 2x y z

3
4
2
 3x y z

3
4 2




3
33
Steps to Simplifying
Expressions
1.
Power of a Quotient Property along with
Power of a Power Property to remove
parenthesis
2. “Flip” negative exponents to make them
positive exponents
3. Use Product of Powers Property
4. Use the Quotient of Powers Property
34
Power of a Quotient
Property and Power of a
Power Property
• Use the power of a quotient property to remove
parenthesis and since the expression has a power
to a power, use the power of a power property.
 2 x 3 y 4 z  2 


3
 3x 4 y 2 z 


3
3
33
43
 23
2 x
y z
 3 43 23 33
3 x y z
35
Continued
• Simplify powers
3
33
43
23
3
9
12
6
2 x
y z
2 x y z
 3 12 6 9
3 43 23 33
3 x y z
3 x y z
36
“Flip” Negative Exponents
to make Positive Exponents
• Now make all of the exponents positive by
looking at each factor and deciding if they
belong in the numerator or denominator.
3
9
12 6
3
9 6 12
6 9
2 x y z
3 x z x y z

3 12 6 9
3 12
3 x y z
2 y
37
Product of Powers
Property
• Now use the product of powers property
to simplify the variables.
3 9 6 12 6 9
912 6 69
3 x z x y z 27 x y z

3 12
12
2 y
6y
21 6 15
27 x y z

12
6y
38
Quotient of Powers
Property
• Now use the Quotient of Powers Property
to simplify.
21 6 15
21 15
21 15
27 x y z
27 x z
27 x z


12
126
6
6y
6y
6y
39
Simplify the Expression
• Simplify:
 5x y z 
  2 3 4 
 2x y z 
3
2 5
4
40
Step 1: Power of a
Quotient Property and
Power of a Power Property
4 12 8 20
5 x y z
 4 8 12 16
2 x y z
41
Step 2: “Flip” Negative
Exponents
4
12 16
2 x z
4 8 20 8 12
5 y z x y
42
Step 3: Product of
Powers Property
4 12 16
2 x z
4 8 20 20
5 x y z
43
Step 4: Quotient of
Powers Property
4
16 x
20 4
625 y z
44
Simplifying Expressions
• Given
4 xy  2 xy


1  3 
2 x y  3 xy
2



2
• Step 1: Power of a
Quotient Property
45
Power of Quotient
Property
• Result after Step 1:
2
2
4
4 xy
2 x y

1 3
2 2 2
2x y
3 x y
• Step 2: Flip Negative Exponents
46
“Flip” Negative
Exponents
3
2
2
2
4 xyxy 3 x y
 2 2 4
2
2 x y
• Step 3: Make one large Fraction by using
the product of Powers Property
47
Make one Fraction by Using
Product of Powers Property
43 x y
3 2 4
2 x y
2
4
6
48
Use Quotient of Powers
Property
2
9x y
2
2
49
Simplify the Expressions
 3a 
1. Try:  1 
 2x 
2
 2x
 4
 y
2
2. Try:
3
 x 
  2 
 4a 



3
3
 2x
 
 y
5
1



2
50
Answers
 3a 
 1 
1. Answer:
 2x 
2
2. Answer:
 2x
 4
 y
2
3
1
 x 
27 a 4 x 6
   2  
2
 4a 



3
3
 2x
 
 y
5



2
2
 4 10
x y
51
Scientific Notation
• Scientific Notation uses powers of ten to express
decimal numbers.
• For example:
2.39  10
5
• The positive exponent means that you move the
decimal to the right 5 times.
•
So,
2.39 10  239,000
5
52
Scientific Notation
• If the exponent of 10 is negative, you
move the decimal to the left the
amount of the exponent.
• Example: 2.65  10
8
 0.0000000265
53
Practice Scientific
Notation
Write the number in
decimal form:
1.
2.
4.9 10
3
1.23 10
6
54
Answers
1.
2.
4.9 10  4,900,000
6
3
1.23 10  0.00123
55
Write a Number in
Scientific Notation
• To write a number in scientific notation, move the
decimal to make a number between 1 and 9.
Multiply by 10 and write the exponent as the
number of places you moved the decimal.
• A positive exponent represents a number larger
than 1 and a negative exponent represents a
number smaller than 1.
56
Example of Writing a
Number in Scientific
Notation
1.
Write 88,000,000 in scientific notation
•
First place the decimal to make a number
between 1 and 9.
Count the number of places you moved the
decimal.
Write the number as a product of the decimal
and 10 with an exponent that represents the
number of decimal places you moved.
Positive exponent represents a number larger
than 1.
•
•
•
8.8  10
7
57
Write 0.0422 in
Scientific Notation
• Move the decimal to make a number between 1 and
9 – between the 4 and 2
• Write the number as a product of the number you
made and 10 to a power 4.2 X 10
• Now the exponent represents the number of
places you moved the decimal, we moved the
decimal 2 times. Since the number is less than 1
the exponent is negative.
4.2  10
2
58
Operations with
Scientific Notation
5
• For example: (2.3 10 )(1.8 10 )
• Multiply 2.3 and 1.8
= 4.14
• Use the product of
powers property 4.14  103 5
• Write in scientific
2
notation 4.14  10
3
59
Try These:
•
Write in scientific notation
1.
(4.110 )(3 10 )
2.
2
5
6
1
(6 10 )(2.5 10 )
60
Answers
1.
2.
(4.110 )(3 10 )  1.23 10
2
5
6
1
(6 10 )(2.5 10 )  1.5 10
5
61
9
The End
• We have completed all the concepts
of simplifying exponents. Now we
just need to practice the concepts!
62