Chapter 8 Exponents and Powers

1
Chapter 8 Exponents and Powers
8-1 11-7 8-2 8-3 Review 8.1 to 8.3
8-4 8-5 8-6 Review 8.4 to 8.6 8-7 8-8 8-9 R
eview Chapter
2
8-1 The Multiplication Counting Principle
8.1

• The student will be able to use the
  Multiplication Counting Principle to determine
  how many choices a situation may give.
• The student will be able to use the Arrangements
  Theorem to determine how many choices a situation
  may give.

3
Multiplication Counting Principle
8.1

• If the first choice can be made in m ways and a
  second choice can be made in n ways, then there
  are mn ways of making the first choice followed
  by the second choice.

4

• Ex 1 Suppose a stadium as 9 gates. Gates A, B, C
  and D are on the north side and gates E, F, G, H
  and I are on the south. In how many ways can you
  enter the stadium through a north gate and leave
  through a south gate?

8.1
Ans 1 We could make an organized list
identifying 1 north gate at a time and cycling
through the south gates
AE, AF, AG, AH, AI, BE, BF, BG, BH, BI CE, CF,
CG, CH, CI, DE, DF, DG, DH, DI Count the number
of combinations we have created. 20 This only
works when there are a small number of
combinations to make.
5
Ex 1 Suppose a stadium as 9 gates. Gates A, B, C
and D are on the north side and gates E, F, G, H
and I are on the south. In how many ways can you
enter the stadium through a north gate and leave
through a south gate?
8.1
Ans 2 Use the Multiplication Counting
Principle The number of choices into the stadium
is 4, while the number of choices out of the
stadium is 5. The total number of ways in a
North gate and out a South gate is
4 5 20
6

• Ex 2 In high school you want to take a
  foreign-language class, a music class and an art
  class. The languages available are French,
  Spanish and Latin. The music classes are band and
  chorus and the art classes are drawing and
  painting. How many different ways can the student
  choose those three classes?

8.1
Step 1 Draw a blank for each decision to be
made _____________ ____________
__________ language choices music
choices art choices Step 2 Put into each
blank the number of ways the
subject could be chosen
3 2
2
Answer there are 3 2 2 12 ways to choose
7
Arrangements Theorem
8.1

• If there are n ways to select each object in a
  sequence of length L, then nL different sequences
  are possible.
• For example A quiz with 20 True-False questions
  would have 220 ways to answer the quiz. 1048576
  different ways to answer the quiz.

8
8.1

• Ex 3 Suppose your next math quiz had 5 questions
  and Student X had not done any homework so had to
  guess. The quiz has two multiple choice questions
  with four choices and three true-false questions.
  A. How many possible ways are there for Student
  X to answer the quiz? B. Whats the
  probability that X will get 100?

4 4 2 2 2
128
different
ways
A. Ans ____ ____ ____ ____ ____ ans
q1 q2 q3 q4 q5
9
8.1

• Ex 4 Ms. Alvarez has written a chapter test. It
  has three multiple-choice questions each with m
  possible answers, two multiple-choice questions
  each with n possible answers, and 5 T-F
  questions. How many ways are there to answer the
  questions?

Ans Question 1 2 3 4 5 6 7 8
9 10 Choices m m m n n 2 2 2
2 2 the number of solutions is
m3n225
32m3n2
10

• Scientific notation a notation used to
  represent very large and very small numbers. The
  number is expressed as a product of a number
  greater than or equal to 1 and less than 10 and a
  power of 10.
• 453.4 4.534 x 10 2
  .00351 3.51 x 10 -3

8.1
11

• Ex 5 Your midterm had 30 four-answer
  multiple-choice questions. How many ways could
  you answer this part of the test?
• State this number rounded to the nearest hundred
  trillion using Scientific Notation.

8.1
430 1,152,921,504,606,846,976
1,152,900,000,000,000,000 1.1529 x 1018
12
11-7 Permutations
11.7

• Objectives
• Find the number of permutations of objects
  without replacement.
• Understand factorial notation.

13
11.7

• Permutation An arrangement where order
  is important. Example P(14,4) 14
  13 12 11
• Factorial n! means the product of all counting
  numbers from n down to 1. Example 6! 6
  5 4 3 2 1 720 P(6,6)
  6!

14
11.7
Ex 1) There are 10 players on a softball team. In
how many ways can the manager choose three
players for first, second, and third base?
Use the Fundamental Counting Principle (8-1)
Permutations of 10 players chosen 3 at a
time P(10,3) Book language
permutations chosen from 10 of length 3
Answer There are 720 different ways the manager
can pick players for first, second, and third
base.
15
11.7
(Your turn) Ex 2) There are 15 students on
student council. In how many ways can Mrs.
Sommers choose three students for president, vice
president, and secretary?
Answer P(15,3) 15 14 13 2,730
Ex 3) Find the value of P(7,5)
7 6 5 4 3 2520
7 things of length 5.
16
11.7
Ex 4) Evaluate 5! Read 5
factorial
5! 5 4 3 2 1 P(5,5)
120
Ex 5) Evaluate

908988 704880
Ex 6) Evaluate


17
11.7
Ex 7) How many ways can you make an 7 digit
number if you only use the digits 1-9
and you must have an even number, and
no number can be used twice?
Digits to be used
1s column digit 2,4,6,8, 4 available
digits
1st digit 1-9 less 1s col digit 8
available digits
2nd digit 1-9 less 2 digits 7 available
digits
This is a permutation for the 1st 6 digits
P(8,6)
7th digit must be even there are 4 digits that
would result in an even
number 2, 4, 6, 8
P(8,6) 4 80,640
18
Definitions Permutations are used to determine
the number of choices when order matters. A
COMBINATION is the number of choices when order
DOESNT matter. C(10,5)

Examples when order doesnt matter Pizza
toppings books in a bag students from a group
(unless 1st vs 2nd)
19
Ex 8 A I want to pick 3 tags from the Xmas
giving tree that has 50 tags on it.
How many ways can I pick the tags? Ex 8 B I
want to pick 3 tags from the Xmas giving tree
that has 50 tags on it. The 1st tag
gets the gift certificate, the 2nd
gets whats in the box and the 3rd gets movie
tickets. How many ways can I pick the
tags?
Combination C(50,3) (504948) / (321)
Permutation P(50,3) 504948
20
8-2 Products and Powers of Powers
8.2

• Objectives
• Simplify products of powers and powers of powers.
• Identify and use the Product of Powers Property
  and the Power of a Power Property.

21
8.2

• Properties
• 1) Product of Powers Property - for all m and n,
  and all nonzero b, bm bn bmn
• 2) Power of a Power Property - for all m and n,
  and all nonzero b, (bm)n bmn

22
8.2

• Examples
• 1) Multiply k4 k2
• 2)A population of guinea pigs is tripling every
  year. If there were 6 guinea pigs at the
  beginning of the year, how many will there be
• a. after 5 years? What kind of relationship is
  this?
• b. four years after that?

k42 k6
633333 6 35 1,458 Exponential growth
6 39 118,098
23
8.2

• 3) Simplify a4 x3 a x10
• 4) Solve for n evaluate (35)3 3n .

a41x310 a5x13
(35)3 35 35 35 3555 315 14348907
n 15
24
8.2

• 5) a. Simplify (x5)2 .
• b. Verify your answer by testing a specific
  case.

x10
Let x 5 (55)2 31252 9,765,625 510
9,765,625
25
8.2

• 6) Simplify 3p2 (p3)4

3p2 p12 3p212 3p14
26
8-3 Quotients of Powers
8.3

• Objectives Evaluate quotients of integer powers
  of real situations
• Simplify quotients of powers
• Identify the Quotient of Powers Property and use
  it to explain operations with powers.
• Use and simplify expressions with quotients of
  powers in real situations.
• Quotient of Powers Property
• Quotient of a Powers Property For all m and n,
  and all nonzero b,
• bm-n

27
8.3

• 1) Change the following to a simplified power
• a. b.
• c.

412-3 49
513-13 50 1
28

• 2) In March, 1992, there was a total of 283.9
  billion dollars in U.S. currency in circulation.
  The U.S. population was about 252.7 million. how
  much currency per person was in circulation?

8.3
1.12347 x 103 1,123.47 per person
29
8.3

• 3) Simplify
• a) b)
• c) d)

4x5
30
Quiz Review 8-1 through 8-3
Rev1

• 1) Fill in the blank with properties of exponents
• (includes the Quotient of a powers property)
• a) bm bn b) (bm)n
• c) bm-n d) 49500000
• 
  scientific notation
• e) 4.394 x 10-6

bmn
bmn
4.95 x 107
bm/bn
0.000004394
31

• 2) Simplify an expression.

Rev1
Simplify coefficients, use properties
Cancel coefficients, use properties
2x4-3y-73z54 2xy-4z9
Simplified expressions have no negative
exponents
32
8-4 Negative Exponents
8.4

• Objectives Evaluate negative integer powers of
  real numbers.
• Simplify products of powers and powers of
  powers involving negative numbers.
• Identify the Negative Exponent Property and use
  it to explain operations with powers.
• Solve problems involving exponential growth and
  decay.
• Use and simplify expressions with powers
  involving negative exponents in real situations.

33
8.4

• Property
• Negative Exponent Property- For any nonzero b
  and all n
• b -n , the reciprocal
  of bn .
• A simplified number or fraction will NOT contain
  any negative exponents.

34
Negative Exponent Property for Fractions
35

• 1) a. Write 7-3 as a simple fraction without a
  negative exponent.
• b. Write as the power of an integer
• (not simplified)
• c. Simplify n5 n-9 , and write your
  answer without negative exponents.

8.4
36-1 or 6-2
36
8.4

• 2) A true-false test has 25 questions. Give the
  probability that you will guess all 25 questions
  correctly.

Answer
37
8.4

• 3) Rewrite (m-5)4 without negative exponents -
  simplify.

Possible Answers
38
8.4

• 4) The viruses in a culture are quadrupling in
  number each day. Right now, the culture contains
  about 1,000,000 viruses. About how many viruses
  did the culture have 6 days ago?

Answer 1,000,000(4)-6 244
39
Application Problem

• Ten years ago, Den put money into a college
  savings account at an annual yield of 7. If the
  money is now worth 9,491.49, what was the amount
  initially invested?
• Hint Use the compound interest formula with a
  negative exponent for the years!
• A P (1r) t

40

• 5)
• a) Rewrite the fraction two different ways
  using positive exponents
• b) Rewrite the fraction two different ways
  using
• negative exponents (not simplified)

8.4
Possible Answers 81-1 9-2 3-4
41
8-5 Powers of Products and Quotients
8.5

• Objectives Evaluate integer powers of real
  number products and quotients.
• Rewrite powers of products and quotients.
• Identify and Power of a product and Power of a
  Quotients Properties and use them to
• explain.
• Use and simplify expression with powers in real
  situations.

42
8.5

• Properties
• 1) Power of Product Property - For all nonzero a
  and b, and for all m
• (ab)m am bm
• 2) Power of a Quotient Property - For all nonzero
  a and b, and for all n

43
8.5

• 1) The length of an edge of one cube is four
  times the length of an edge or another cube. The
  volume of the larger cube is how many times the
  volume of the smaller cube?

Answer Vsmaller s3 Vlarger (4s)3 ?
Vol of larger is 43 (64) times Vol of smaller.
44

• 2) The sun's radius is about 6.96 105 km.
  Estimate the volume of the sun. Volume of a
  sphere pr3

8.5
1412.27 x 1015
1.41227 x 1018 simplify to scientific
notation
45
8.5

• 3) Simplify -(-a2 b3)4

-a8 b12
Definition of simplified no parenthesis and
no negative exponents
46
8.5

• 4) Write as a simple fraction

47
8.5
5) Rewrite as a single
fraction
48
8-6 Square Roots and Cube Roots
8.6

• Objectives
• Day 1
• Evaluate Squares and Square roots
• Use the Pythagorean Theorem.
• Day 2
• Evaluate Cubes and Cube roots.
• Review

49
8.6 D1

• Definitions
• If A s2, then s (square root of A)
• The Radical sign is the square root symbol. The
  horizontal (top) bar works as a set of
  parenthesis. For example
• 32 42 h2
• h
• You must first square the 3 and the 4,
  then add before taking the square root.

50
8.6 D1

• Definitions
• Square of the square root property For any
  nonnegative x
• The square root is really the ½ power.

Powers of Products Rule
51
8.6 D1

• 1) Evaluate

52
8.6 D1

• 2 a) Estimate to the nearest whole
  number.

9 lt 12 lt 16 32 9 42 16
9 3 12 and 16 - 12 4 9 is closer (by 1)
Answer The nearest whole number is 3, but it
will be between 3 4.
2 b) Estimate to two decimal places.
Use a calculator 3.46
53
8.6 D1

• 3) Evaluate without a calculator

Remember
4 10
Answer 40
4) Solve x2 81
x 9
54
8.6 D1
5) Solve for the missing side to the nearest
tenth.

10ft a) 7cm
b) 7ft 5cm
102 72 c2
52 b2 72 b2 72 - 52
12.2ft c
b 4.9cm
55
8.6 D2

• Definitions
• If V s3, then s is a cube root of V.
• Example 64 43 , then 4 is the cube root of
  64.
• Cube of the Cube Root Property
• For any nonnegative number x,
• Note

56
8.6 D2

• 1) Find the cube root of 27
• 2) Find the cube root of 81 to the nearest tenth
• 3) If the volume of a cube is 200cm3 , find the
  length of the side of the cube to the nearest
  hundredth.

3 3 3 27 so the cube root of 27 3
On a calculator type 81(1/3) 4.3
s3 200
s 5.85
Check 5.853 200
57
8.6
4) Simplify 5 3 5) Fill in
the blanks Since 53 125, ______ is
the cube root of _________. 6)
List all the perfect cubes up to 1000
11.31 approximate answer
5
125
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
58

• More with Square Roots
• 7)
• 8)
• 9) 5(2 3 )

8.6
53 24 23
59
8.4 8.6 Review
Rev 2

• Simplify
• 1) 6)
  The principal square
• 
  root is _________?
• 2) (k-3)4 7) Give
  exact and approx
• square roots of 15.
• 3) (-2y)4
• 4) (-4x3yz4)5
• 8)
  If f(y) 3 find f(8).
• 5)

positive
16y4
-1024x15y5z20
24
60
8.4 8.6 Review
Rev 2

• Write an exact value and a value approximated to
  the nearest hundredth.
• 9)
  
• 10)
• 
  13) solve for x
• 11)
  
• 12)

(hint use a negative exponent!)
25
x -3
61
8-7 Multiplying and Dividing Square Roots
8.7

• Objectives Simplify square roots.
• Property
• Products and Quotients of Square Roots for all
  nonnegative real numbers a, b and c,

The square root ofany number that isnot a
perfect squareis an irrational number.
62

• 1) Verify that
• a. by finding decimal approximations.
• b. by squaring each side.

8.7
Property of Equality
50 10 5 50v
63

• 2) Simplify

8.7
Step 1 Find the largest perfect square that is a
factor of the number under the
radical
Step 2 Break the radical into pieces where the
perfect square from step one is by
itself.
Step 3 Simplify
64

• 3) Find the exact length of the hypotenuse of a
  right triangle with legs of 6 and 4.

8.7
a2 b2 c2 62 42 c2 52 c2
Pythagorean Theorem
6 4
Substitution
Simplify the right side
c
Property of Equality
c
Simplify the radical
c
65

• 4) Simplify

8.7
Find factors of the radical
Simplify the radical
UNdistribute 4 from each termand cancel.
66

• 5) Assume a and b are positive. Find
  and simplify the result.

8.7
Method 1
Method 2
67

• 6) Assume a and b are positive. Simplify

8.7
7) Given n 0, simplify
8) Simplify
68

• 9) Solve y2 9 33 and simplify the answer.

8.7
y2 24
10) Simplify 11)
69
(No Transcript)
70
8-8 Distance in a Plane
8.8

• Objectives Use the Pythagorean Theorem and
  simplify the radical answers to get exact values.
• Distance Formula
• The distance between A(x1,y1) and B (x2,y2)
• in a coordinate plane

71

• 1) Refer to the map of NYC. How far is it from
  the intersection of 44 St and 7th Ave to the
  intersection of 34th and 8th Ave. (long blocks
  1056 ft, short 264 ft)
• a. down 7th and over 34th?
• b. traveling along Broadway (as the crow flies)?

8.8
10(264) 1(1056) 3696 ft
NY_at_tlas Stephan Van Dam
72

• 2) Triangle LMN has coordinates L (-6, -2), M
  (6, 4), N (6, -2) . Find the length of MN as a
  simplified radical. Draw the triangle on a grid
  if necessary.

8.8
c2 a2 b2 c
Pythagorean Theorem
Formula
Distance y2 y1
Distance x2 x1
Substitution
Simplify whats under the radical
Find the perfect square factor
Answer
Simplify the radical
73

• 3) Find the exact distance between (-11, 2) and
  (-6, 7).

8.8
Answer
4) Give the distance as a simplified radical and
a decimal. a. Let C (4,2) and K (7,11).
Find CK. b. Let N (-5, 4) and Q (2, -2).
Find NQ.
9.49


9.22

74

• 5) Tony and Alicia each left camp on snowmobiles.
  Tony drove one mile north, then 5 miles west.
  Alicia drove 6 miles east, then 2 miles south.
  Make a diagram and find the distance between Tony
  and Alicia.

8.8
11.4 miles
Think If camp is (0,0)what are their
coordinateswhen they stop?
75

• Extra Practice
• 6)
• 7)
• 8)

8.8
76
8-9 Remembering Properties of Exponents and
Powers
8.9

• Objectives Evaluate integer powers of real
  numbers.
• Test a special case to determine whether a
  pattern is true.

77
8.9

• 1) Write as a simple fraction.

2) Simplify x -4 x10 Test a special case
to check your answer.
x6
Let x 3 then 3-4 310 1/81
59049 729 36
78
8.9

• 3) Use a counter example to show that
  ?

5 ? 7
79
8.9

• 4) Simplify
• 5) Tell whether the pattern
  is true.
• 6) Simplify to an exact value
  .

80
8.9

• 7) Simplify
• 8) Simply
• 9) Simplify

3
81
Chapter Review
Chpt Rev

• 1) Simplify
• a. Product of Powers Property bm bn
  bmn
• x2 x-2
• (y2 y4 )y3

1
y9
82
Chpt Rev

• b. Power of a Power Property (bm)n bmn
• Power of a Product Property (ab)n an bn
• (2x3 y2)4

24x12y8 16x12y8
c. Negative Exponent Property b-n 4-3

32
53
83
Chpt Rev

• d. Quotient of a Powers Property

74
e. Power of a quotient Property
84

• Simplify the exponents and negatives
• (M4)-3
• (-32n-5)2
• -(-50(p-2)2)-3

Chpt Rev
8.4
M-12 1/M12
34n-10 34 / n10
p12
85

• 6) Seven Chicago Bears football fans each wrote a
  letter from the phrase GO BEARS on their chests
  to show their team support. When they arrived at
  the game they sat next to each other in random
  order.
• How many different forms of the phrase are
  possible?
• Write your answer in scientific notation.
• What is the probability that the fans spelled the
  phrase correctly when they first sat down?

Chpt Rev
7! 7654321 5040
5040 5.04 x 103
86
Chpt Rev

• 7) Simplify
• (-7a)3 -7a3 (-7a)2 -72a

-343a3
-7a3
49a2
-49a
8) Find a counterexample. Define it ex (xy)2
? x2 y2
Let x 3 y 4 (34)2 ? 32 44 49 ? 25
87

• 9) Evaluate an expression.
• 10) Rewrite without negative exponents.
• 4r-3 s2 t-1