1. 1
SimplifyingSimplifying
ExponentsExponents
SimplifyingSimplifying
ExponentsExponents
Algebra IAlgebra I

2. 2
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

3. 3
Multiplication Properties
of Exponents
•
• Product of Powers Property
• Power of a Power Property
• Power of a Product Property

4. 4
Product of Powers
Property
•
• To multiply powers
that have the same
base, you add the
exponents.
•
•
• Example: 53232
aaaaaaaaa ==⋅⋅⋅⋅=⋅ +

5. 5
Practice Product of
Powers Property:
•
•
• Try:
•
•
• Try:
325
nnn ⋅⋅
45
xx ⋅

6. 6
Answers To Practice
Problems
•
1. Answer:
2.
3.
4. Answer:
94545
xxxx ==⋅ +
10325325
nnnnn ==⋅⋅ ++

7. 7
Power of a Power
Property
• To find a power of a power, you
multiply the exponents.
•
• Example:
• Therefore,
622222232
)( aaaaaa ==⋅⋅= ++
63232
)( aaa == ⋅

8. 8
Practice Using the Power
of a Power Property
1.
2. Try:
3.
4.
5. Try:
44
)( p
54
)(n

9. 9
Answers to Practice
Problems
•
1. Answer:
2.
3.
4. Answer:
164444
)( ppp == ⋅
205454
)( nnn == ⋅

10. 10
Power of a Product
Property
•
• To find a power of a product, find
the power of EACH factor and
multiply.
•
• Example:
333333
644)4( zyzyyz =⋅⋅=

11. 11
Practice Power of a
Product Property
•
1. Try:
2.
3.
4.
5. Try:
6
)2( mn
4
)(abc

12. 12
Answers to Practice
Problems
•
1. Answer:
2.
3.
4. Answer:
666666
642)2( nmnmmn ==
4444
)( cbaabc =

13. 13
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.

14. 14
Zero Exponents
• Any number, besides zero, to the
zero power is 1.
•
• Example:
•
• Example:
10
=a
140
=

15. 15
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.

16. 16
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.
3
3 1
x
x =−

17. 17
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
4
4
1
1
y
y
y
==−

18. 18
Practice Making Negative
Exponents Positive
1. Try:
2.
3.
4.
5. Try:
3−
d
5
1
−
z

19. 19
Answers to Negative
Exponents Practice
•
1. Answer:
2.
3.
4. Answer:
3
3 1
d
d =−
5
5
5
1
1
z
z
z
==−

20. 20
Rewrite the Expression
with Positive Exponents
•
• Example:
•
• 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.
•
23
2 −−
yx
xy
yx
2
2 23
=−−

21. 21
Rewrite the Expression
with Positive Exponents
•
• Example:
•
•
• 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.
833
4 −−
cab

22. 22
Rewriting the Expression
with Positive Exponents
•
• 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.
•
833
4 −−
cab
8
3
83
3
644 c
ab
c
ab
=

23. 23
Practice Rewriting the
Expressions with Positive
Exponents:
1.
2.
3.
4. Try:
5.
6.
7. Try:
zyx 321
3 −−−
dcba 432
4 −−

24. 24
Answers
1.
2.
3. Answer
4.
5.
6. Answer
32
321
3
3
yx
z
zyx =−−−
42
3
432 4
4
ca
db
dcba =−−

25. 25
Division Properties of
Exponents
•
• Quotient of Powers Property
•
• Power of a Quotient Property

26. 26
Quotient of Powers
Property
•
• To divide powers
that have the
same base,
subtract the
exponents.
•
• Example:
2
35
3
5
1
x
x
x
x
==
−

27. 27
Practice Quotient of
Powers Property
•
1. Try:
2.
3.
4.
5. Try:
3
9
a
a
4
3
y
y

28. 28
Answers
•
1. Answer:
2.
3.
4.
5. Answer:
6
39
3
9
1
a
a
a
a
==
−
yyy
y 11
344
3
== −

29. 29
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: 3
33
b
a
b
a
=






30. 30
Simplifying Expressions
•
•
•
• Simplify
343
3
2






mn
nm

31. 31
Simplifying Expressions
•
• First use the Power of a Quotient Property
along with the Power of a Power Property
333
1293
333
34333343
3
2
3
2
3
2
nm
nm
nm
nm
mn
nm
==




 ⋅⋅

32. 32
Simplify Expressions
•
• Now use the
Quotient of
Power Property
•
27
8
27
8
3
2 9631239
333
1293
nmnm
nm
nm
==
−−

33. 33
Simplify Expressions
•
• Simplify 3
24
243
3
3
2
−
−−








zyx
zyx

34. 34
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
5.
6.
7.

35. 35
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.
•
•
3332343
3234333
3
24
243
3
2
3
2
3 −⋅−⋅−⋅−
−⋅−−⋅−⋅−−
−
−−
=







zyx
zyx
zyx
zyx

36. 36
Continued
•
• Simplify powers
•
•
96123
61293
3332343
3234333
3
2
3
2
−−−−
−−
−⋅−⋅−⋅−
−⋅−−⋅−⋅−−
=
zyx
zyx
zyx
zyx

37. 37
“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.
•
•
123
9612693
96123
61293
2
3
3
2
y
zyxzx
zyx
zyx
=−−−−
−−

38. 38
Product of Powers
Property
•
• Now use the product of powers property
to simplify the variables.
•
•
12
15621
12
966129
123
9612693
6
27
6
27
2
3
y
zyx
y
zyx
y
zyxzx
==
++

39. 39
Quotient of Powers
Property
•
• Now use the Quotient of Powers Property
to simplify.
•
•
6
1521
612
1521
12
15621
6
27
6
27
6
27
y
zx
y
zx
y
zyx
== −

40. 40
Simplify the Expression
•
• Simplify:
4
432
523
2
5
−
−−
−






zyx
zyx

41. 41
Step 1: Power of a
Quotient Property and
Power of a Power Property•
•
•
161284
208124
2
5
−−
−−−
zyx
zyx

42. 42
Step 2: “Flip” Negative
Exponents
1282084
16124
5
2
yxzy
zx
•
•

43. 43
Step 3: Product of
Powers Property
•
•
•
202084
16124
5
2
zyx
zx

44. 44
Step 4: Quotient of
Powers Property
420
4
625
16
zy
x
•
•
•

45. 45
Simplifying Expressions
•
• Given
•
•
• Step 1: Power of a
Quotient
Property
22
31
3
2
2
4
−
−− 





⋅
xy
xy
yx
xy

46. 46
Power of Quotient
Property
• Result after Step 1:
•
•
•
•
• Step 2: Flip Negative Exponents
222
422
31
3
2
2
4
−−−
−−−
−−
⋅
yx
yx
yx
xy

47. 47
“Flip” Negative
Exponents
•
•
•
• Step 3: Make one large Fraction by using
the product of Powers Property
422
2223
2
3
2
4
yx
yxxyxy
⋅

48. 48
Make one Fraction by Using
Product of Powers Property
423
642
2
34
yx
yx⋅
•
•
•

49. 49
Use Quotient of Powers
Property
2
9 22
yx
•
•
•

50. 50
Simplify the Expressions
•
1.
2. Try:
3.
4.
5.
6. Try:
1
2
33
1
2
42
3
−
−
−
− 





⋅





a
x
x
a
253
4
2
22
−






⋅





y
x
y
x

51. 51
Answers
•
•
1. Answer:
2.
3.
4. Answer:
2
27
42
3 641
2
33
1
2
xa
a
x
x
a
=





⋅





−
−
−
−
104
253
4
2
222
yxy
x
y
x
=





⋅





−

52. 52
Scientific Notation
• Scientific Notation uses powers of ten to express
decimal numbers.
•
• For example:
•
• The positive exponent means that you move the
decimal to the right 5 times.
• So,
•
5
1039.2 ×
000,2391039.2 5
=×

53. 53
Scientific Notation
•
• If the exponent of 10 is negative,
you move the decimal to the left
the amount of the exponent.
•
• Example: 0000000265.01065.2 8
=× −

54. 54
Practice Scientific
Notation
•
Write the number in
decimal form:
1.
2.
1.
6
109.4 ×
3
1023.1 −
×

55. 55
Answers
•
•
1.
2.
000,900,4109.4 6
=×
00123.01023.1 3
=× −

56. 56
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.

57. 57
Example of Writing a
Number in Scientific
Notation1.
2. 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.
1. 7
108.8 ×

58. 58
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.
2
102.4 −
×

59. 59
Operations with
Scientific Notation
• For example:
• Multiply 2.3 and 1.8
= 4.14
• Use the product of
powers property
• Write in scientific
notation
)108.1)(103.2( 53 −
××
53
1014.4 −+
×
2
1014.4 −
×

60. 60
Try These:
• Write in scientific notation
1.
2.
)103)(101.4( 62
××
)105.2)(106( 15 −−
××

61. 61
Answers
•
•
1.
2.
962
1023.1)103)(101.4( ×=××
515
105.1)105.2)(106( ×=×× −−

62. 62
The End
• We have completed all the concepts
of simplifying exponents. Now we
just need to practice the concepts!