0
1
SimplifyingSimplifying
ExponentsExponents
SimplifyingSimplifying
ExponentsExponents
Algebra IAlgebra I
2
Contents
• Multiplication Properties of Exponents ……….1 – 13
• Zero Exponent and Negative Exponents……14 – 24
• Division ...
3
Multiplication Properties
of Exponents
•
• Product of Powers Property
• Power of a Power Property
• Power of a Product P...
4
Product of Powers
Property
•
• To multiply powers
that have the same
base, you add the
exponents.
•
•
• Example: 53232
a...
5
Practice Product of
Powers Property:
•
•
• Try:
•
•
• Try:
325
nnn ⋅⋅
45
xx ⋅
6
Answers To Practice
Problems
•
1. Answer:
2.
3.
4. Answer:
94545
xxxx ==⋅ +
10325325
nnnnn ==⋅⋅ ++
7
Power of a Power
Property
• To find a power of a power, you
multiply the exponents.
•
• Example:
• Therefore,
622222232
...
8
Practice Using the Power
of a Power Property
1.
2. Try:
3.
4.
5. Try:
44
)( p
54
)(n
9
Answers to Practice
Problems
•
1. Answer:
2.
3.
4. Answer:
164444
)( ppp == ⋅
205454
)( nnn == ⋅
10
Power of a Product
Property
•
• To find a power of a product, find
the power of EACH factor and
multiply.
•
• Example:
...
11
Practice Power of a
Product Property
•
1. Try:
2.
3.
4.
5. Try:
6
)2( mn
4
)(abc
12
Answers to Practice
Problems
•
1. Answer:
2.
3.
4. Answer:
666666
642)2( nmnmmn ==
4444
)( cbaabc =
13
Review Multiplication
Properties of Exponents
•
• Product of Powers Property—To multiply powers
that have the same base...
14
Zero Exponents
• Any number, besides zero, to the
zero power is 1.
•
• Example:
•
• Example:
10
=a
140
=
15
Negative Exponents
•
• To make a negative
exponent a positive
exponent, write it as
its reciprocal.
• In other words, w...
16
Negative Exponent
Examples
•
• Example of Negative
Exponent in the
Numerator:
•
• The negative exponent
is in the numer...
17
Negative Exponents
Example
•
• Negative Exponent in the
Denominator:
•
• The negative exponent is in
the denominator, s...
18
Practice Making Negative
Exponents Positive
1. Try:
2.
3.
4.
5. Try:
3−
d
5
1
−
z
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
Rewrite the Expression
with Positive Exponents
•
• Example:
•
• Look at EACH factor and decide if the factor belongs in...
21
Rewrite the Expression
with Positive Exponents
•
• Example:
•
•
• All the factors are in the numerator. Now
look at eac...
22
Rewriting the Expression
with Positive Exponents
•
• Example:
• The 4 has a negative exponent so to make the exponent p...
23
Practice Rewriting the
Expressions with Positive
Exponents:
1.
2.
3.
4. Try:
5.
6.
7. Try:
zyx 321
3 −−−
dcba 432
4 −−
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
Division Properties of
Exponents
•
• Quotient of Powers Property
•
• Power of a Quotient Property
26
Quotient of Powers
Property
•
• To divide powers
that have the
same base,
subtract the
exponents.
•
• Example:
2
35
3
5...
27
Practice Quotient of
Powers Property
•
1. Try:
2.
3.
4.
5. Try:
3
9
a
a
4
3
y
y
28
Answers
•
1. Answer:
2.
3.
4.
5. Answer:
6
39
3
9
1
a
a
a
a
==
−
yyy
y 11
344
3
== −
29
Power of a Quotient
Property
•
• To find a power of a
quotient, find the
power of the
numerator and the
power of the
de...
30
Simplifying Expressions
•
•
•
• Simplify
343
3
2






mn
nm
31
Simplifying Expressions
•
• First use the Power of a Quotient Property
along with the Power of a Power Property
333
129...
32
Simplify Expressions
•
• Now use the
Quotient of
Power Property
•
27
8
27
8
3
2 9631239
333
1293
nmnm
nm
nm
==
−−
33
Simplify Expressions
•
• Simplify 3
24
243
3
3
2
−
−−








zyx
zyx
34
Steps to Simplifying
Expressions
1. Power of a Quotient Property along with
Power of a Power Property to remove
parenth...
35
Power of a Quotient
Property and Power of a
Power Property•
• Use the power of a quotient property to remove
parenthesi...
36
Continued
•
• Simplify powers
•
•
96123
61293
3332343
3234333
3
2
3
2
−−−−
−−
−⋅−⋅−⋅−
−⋅−−⋅−⋅−−
=
zyx
zyx
zyx
zyx
37
“Flip” Negative Exponents
to make Positive Exponents
•
• Now make all of the exponents positive by
looking at each fact...
38
Product of Powers
Property
•
• Now use the product of powers property
to simplify the variables.
•
•
12
15621
12
966129...
39
Quotient of Powers
Property
•
• Now use the Quotient of Powers Property
to simplify.
•
•
6
1521
612
1521
12
15621
6
27
...
40
Simplify the Expression
•
• Simplify:
4
432
523
2
5
−
−−
−






zyx
zyx
41
Step 1: Power of a
Quotient Property and
Power of a Power Property•
•
•
161284
208124
2
5
−−
−−−
zyx
zyx
42
Step 2: “Flip” Negative
Exponents
1282084
16124
5
2
yxzy
zx
•
•
43
Step 3: Product of
Powers Property
•
•
•
202084
16124
5
2
zyx
zx
44
Step 4: Quotient of
Powers Property
420
4
625
16
zy
x
•
•
•
45
Simplifying Expressions
•
• Given
•
•
• Step 1: Power of a
Quotient
Property
22
31
3
2
2
4
−
−− 





⋅
xy
xy
y...
46
Power of Quotient
Property
• Result after Step 1:
•
•
•
•
• Step 2: Flip Negative Exponents
222
422
31
3
2
2
4
−−−
−−−
...
47
“Flip” Negative
Exponents
•
•
•
• Step 3: Make one large Fraction by using
the product of Powers Property
422
2223
2
3
...
48
Make one Fraction by Using
Product of Powers Property
423
642
2
34
yx
yx⋅
•
•
•
49
Use Quotient of Powers
Property
2
9 22
yx
•
•
•
50
Simplify the Expressions
•
1.
2. Try:
3.
4.
5.
6. Try:
1
2
33
1
2
42
3
−
−
−
− 





⋅





a
x
x
a
253...
51
Answers
•
•
1. Answer:
2.
3.
4. Answer:
2
27
42
3 641
2
33
1
2
xa
a
x
x
a
=





⋅





−
−
−
−
104
253...
52
Scientific Notation
• Scientific Notation uses powers of ten to express
decimal numbers.
•
• For example:
•
• The posit...
53
Scientific Notation
•
• If the exponent of 10 is negative,
you move the decimal to the left
the amount of the exponent....
54
Practice Scientific
Notation
•
Write the number in
decimal form:
1.
2.
1.
6
109.4 ×
3
1023.1 −
×
55
Answers
•
•
1.
2.
000,900,4109.4 6
=×
00123.01023.1 3
=× −
56
Write a Number in
Scientific Notation
•
•
• To write a number in scientific notation, move the
decimal to make a number...
57
Example of Writing a
Number in Scientific
Notation1.
2. Write 88,000,000 in scientific notation
• First place the decim...
58
Write 0.0422 in
Scientific Notation
•
• Move the decimal to make a number between 1
and 9 – between the 4 and 2
• Write...
59
Operations with
Scientific Notation
• For example:
• Multiply 2.3 and 1.8
= 4.14
• Use the product of
powers property
•...
60
Try These:
• Write in scientific notation
1.
2.
)103)(101.4( 62
××
)105.2)(106( 15 −−
××
61
Answers
•
•
1.
2.
962
1023.1)103)(101.4( ×=××
515
105.1)105.2)(106( ×=×× −−
62
The End
• We have completed all the concepts
of simplifying exponents. Now we
just need to practice the concepts!
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Transcript of "Simplifying exponents"

  1. 1. 1 SimplifyingSimplifying ExponentsExponents SimplifyingSimplifying ExponentsExponents Algebra IAlgebra I
  2. 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. 3 Multiplication Properties of Exponents • • Product of Powers Property • Power of a Power Property • Power of a Product Property
  4. 4. 4 Product of Powers Property • • To multiply powers that have the same base, you add the exponents. • • • Example: 53232 aaaaaaaaa ==⋅⋅⋅⋅=⋅ +
  5. 5. 5 Practice Product of Powers Property: • • • Try: • • • Try: 325 nnn ⋅⋅ 45 xx ⋅
  6. 6. 6 Answers To Practice Problems • 1. Answer: 2. 3. 4. Answer: 94545 xxxx ==⋅ + 10325325 nnnnn ==⋅⋅ ++
  7. 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. 8 Practice Using the Power of a Power Property 1. 2. Try: 3. 4. 5. Try: 44 )( p 54 )(n
  9. 9. 9 Answers to Practice Problems • 1. Answer: 2. 3. 4. Answer: 164444 )( ppp == ⋅ 205454 )( nnn == ⋅
  10. 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. 11 Practice Power of a Product Property • 1. Try: 2. 3. 4. 5. Try: 6 )2( mn 4 )(abc
  12. 12. 12 Answers to Practice Problems • 1. Answer: 2. 3. 4. Answer: 666666 642)2( nmnmmn == 4444 )( cbaabc =
  13. 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. 14 Zero Exponents • Any number, besides zero, to the zero power is 1. • • Example: • • Example: 10 =a 140 =
  15. 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. 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. 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. 18 Practice Making Negative Exponents Positive 1. Try: 2. 3. 4. 5. Try: 3− d 5 1 − z
  19. 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. 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. 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. 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. 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. 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. 25 Division Properties of Exponents • • Quotient of Powers Property • • Power of a Quotient Property
  26. 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. 27 Practice Quotient of Powers Property • 1. Try: 2. 3. 4. 5. Try: 3 9 a a 4 3 y y
  28. 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. 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. 30 Simplifying Expressions • • • • Simplify 343 3 2       mn nm
  31. 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. 32 Simplify Expressions • • Now use the Quotient of Power Property • 27 8 27 8 3 2 9631239 333 1293 nmnm nm nm == −−
  33. 33. 33 Simplify Expressions • • Simplify 3 24 243 3 3 2 − −−         zyx zyx
  34. 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. 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. 36 Continued • • Simplify powers • • 96123 61293 3332343 3234333 3 2 3 2 −−−− −− −⋅−⋅−⋅− −⋅−−⋅−⋅−− = zyx zyx zyx zyx
  37. 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. 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. 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. 40 Simplify the Expression • • Simplify: 4 432 523 2 5 − −− −       zyx zyx
  41. 41. 41 Step 1: Power of a Quotient Property and Power of a Power Property• • • 161284 208124 2 5 −− −−− zyx zyx
  42. 42. 42 Step 2: “Flip” Negative Exponents 1282084 16124 5 2 yxzy zx • •
  43. 43. 43 Step 3: Product of Powers Property • • • 202084 16124 5 2 zyx zx
  44. 44. 44 Step 4: Quotient of Powers Property 420 4 625 16 zy x • • •
  45. 45. 45 Simplifying Expressions • • Given • • • Step 1: Power of a Quotient Property 22 31 3 2 2 4 − −−       ⋅ xy xy yx xy
  46. 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. 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. 48 Make one Fraction by Using Product of Powers Property 423 642 2 34 yx yx⋅ • • •
  49. 49. 49 Use Quotient of Powers Property 2 9 22 yx • • •
  50. 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. 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. 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. 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. 54 Practice Scientific Notation • Write the number in decimal form: 1. 2. 1. 6 109.4 × 3 1023.1 − ×
  55. 55. 55 Answers • • 1. 2. 000,900,4109.4 6 =× 00123.01023.1 3 =× −
  56. 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. 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. 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. 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. 60 Try These: • Write in scientific notation 1. 2. )103)(101.4( 62 ×× )105.2)(106( 15 −− ××
  61. 61. 61 Answers • • 1. 2. 962 1023.1)103)(101.4( ×=×× 515 105.1)105.2)(106( ×=×× −−
  62. 62. 62 The End • We have completed all the concepts of simplifying exponents. Now we just need to practice the concepts!
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