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 8.68-7 8-8 8-9 Review Chapter

8-1 The Multiplication Counting Principle

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.

8.1

Multiplication Counting Principle

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

8.1

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?

AE, AF, AG, AH, AI, BE, BF, BG, BH, BICE, 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.

Ans 1: We could make an organized list identifying 1 north gate at a time and cycling through the south gates:

8.1

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?

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

8.1

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?

Step 1: Draw a blank for each decision to be made… _____________ ____________ __________ language choices music choices art choicesStep 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

8.1

Arrangements Theorem

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.

8.1

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. What’s the probability that X will get 100%?

A. Ans: ____ ____ ____ ____ ____# ans: q1 q2 q3 q4 q5

4 · 4 · 2 · 2 · 2 = 128 different ways

B. Ans – exactly 1 set of choices is 100% correct: the probability is: 1

128

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

8.1

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

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.

430 = 1,152,921,504,606,846,976

1,152,900,000,000,000,0001.1529 x 1018

8.1

11-7 Permutations

• Objectives :

• Find the number of permutations of objects without replacement.

• Understand factorial notation.

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!

11.7

Number of items to multiplyValue to start with

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?

10 9 8 720x x =

number of possible players for first base

number of possible

players for second base

number of possible

players for third base

total number of possible

waysx x =

Answer: There are 720 different ways the manager can pick players for first, second, and third base.

= Permutations of 10 players chosen 3 at a time= P(10,3)

Book language: permutations chosen from 10 of length 3

11.7

Use the Fundamental Counting Principle (8-1)

(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

7 things of length 5.

Ex 3) Find the value of P(7,5)

7 · 6 · 5 · 4 · 3 = 2520

11.7

90!

87!

21!

25 24 23 22 21!

90 89 88 87!

87!

Ex 4) Evaluate 5! Read “5 factorial”

5! = 5 · 4 · 3 · 2 · 1 = P(5,5)

= 120

Ex 5) Evaluate = 90·89·88 = 704880=

Ex 6) Evaluate 21!

25!= =

1

25 24 23 2

1

3036002

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:

1’s column digit: 2,4,6,8, 4 available digits

1st digit: 1-9 less 1’s col digit 8 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

11.7

2nd digit: 1-9 less 2 digits 7 available digits …

Definitions: Permutations are used to determine the number of choices when order matters. A COMBINATION is the number of choices when order DOESN’T matter.

C(10,5) = (10,5)

5!

P= 10 9 8 7 6 3 2 7 6

5 4 3 2 1 1

Examples when order doesn’t matter: Pizza toppingsbooks in a bagstudents from a group (unless 1st vs 2nd)

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 what’s in the box and the 3rd gets movie tickets. How many ways can I pick the tags?

Permutation = P(50,3) = 50∙49∙48

Combination = C(50,3) = (50∙49∙48) / (3∙2∙1)

8-2 Products and Powers of Powers

Objectives:

Simplify products of powers and powers of powers.

Identify and use the Product of Powers Property and the Power of a Power Property.

8.2

Properties

1) Product of Powers Property - for all m and n, and all nonzero b, bm ∙ bn = bm+n

2) Power of a Power Property - for all m and n, and all nonzero b, (bm)n = bmn

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?

k4+2 = k6

6·3·3·3·3·3 = 6 · 35 = 1,458; Exponential growth

= 6 · 39 = 118,098

8.2

3) Simplify a4 ∙ x3 ∙ a ∙ x10

4) Solve for n & evaluate (35)3 = 3n .

= a4+1x3+10 = a5x13

= (35)3 = 35 · 35 · 35 = 35+5+5

= 315 = 14348907 n = 15

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

8.2

6) Simplify 3p2 ∙ (p3)4

3p2 · p12 = 3p2+12 = 3p14

8.2

8-3 Quotients of PowersObjectives: Evaluate quotients of integer powers of real situationsSimplify quotients of powersIdentify 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 PropertyQuotient of a Powers Property: For all m and n, and all

nonzero b,

= bm-nm

n

b

b

8.3

1) Change the following to a simplified power:

a. b.

c.

12

3

4

4

3

12

4

4

13

13

5

5

412-3 = 49 43-12 = 4-9 =9

1

4

513-13 = 50 = 1

8.3

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?

11 11

8 8

283,900,000,000 2.839 10 2.839 10

252,700,000 2.527 10 2.527 10

≈ 1.12347 x 103 ≈ $1,123.47 per person

8.3

3) Simplify

a) b)

c) d)

2 10

15

30

25

x y

xy

7

2

12

3

x

x

14

17

28x10

7x10

5

8

a

a

5

6

5

x

y = 4x5

4·10-3=

1 44

1000 1000.004

3

1

a

8.3

Quiz Review: 8-1 through 8-3

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 =

bm+n bm·n

bm/bn 4.95 x 107

0.000004394

Rev1

2) Simplify an expression.

4 5 1 3

2 4 2 3

56 4

16 7

x y xz x y z

xy z x z

Simplify coefficients, use properties

= 2x4-3y-7+3z5+4 = 2xy-4z9 =

Cancel coefficients, use properties

7

2x4+1-1y-5-2z1-(-4)

4

7x-1-2y3z1-(-3)=

Rev1

9

4

2xz

ySimplified expressions have no negative exponents

8-4 Negative Exponents

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.

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.

1nb

8.4

Negative Exponent Property for Fractions

2 25 4

4 5

n nx y

y x

33 26

2

1

1

mm

m

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.

1

36

8.4

3

1

7

36-1 or 6-2

4

1

n

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

8.4

Answer: 8

25

1 13 10 0%

2 33554432x

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

8.4

Possible Answers: 20

1

m

4

5

1

m

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?

8.4

Answer: 1,000,000(4)-6 ≈ 244

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 (1+r) t

5)

a) Rewrite the fraction two different ways using positive exponents:

b) Rewrite the fraction two different ways using

negative exponents (not simplified):

1

81

8.4

1

81

Possible Answers: 81-1 9-2 3-4

4

1

32

1

9Answer:

8-5 Powers of Products and Quotients

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.

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

n n

n

a a

b b

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?

8.5

Answer: Vsmaller = s3; Vlarger= (4s)3 Vol of larger is 43 (64) times Vol of smaller.

2) The sun's radius is about 6.96 ∙ 105 km. Estimate the volume of the sun. Volume of a sphere = πr34

3

8.5

4

3π(6.96x105)3

4

3π(6.96)3 x (10)15

≈ 1412.27 x 1015

≈ 1.41227 x 1018 simplify to scientific notation

3) Simplify -(-a2 b3)48.5

= -a8 b12

Definition of simplified: no parenthesis; and no negative exponents

4) Write as a simple fraction4

4

5

8.5

256

625=

n n

n

a a

b b

Remember:

32 7

4

x

y xy

5) Rewrite as a single fraction

8.5

3

2 4

7

2x y

8-6 Square Roots and Cube Roots

Objectives:

Day 1

Evaluate Squares and Square roots

Use the Pythagorean Theorem.

Day 2

Evaluate Cubes and Cube roots.

Review

8.6

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.

A

8.6 D1

2 23 4

Definitions:

Square of the square root property: For any nonnegative x:

The square root is really the ½ power.

2x x x x

8.6 D1

1 1 1 112 2 2 2x x x x x x x

Powers of Products Rule

1) Evaluate2 212 5

8.6 D1

13Answer:

144 25

169

2 a) Estimate to the nearest whole number.128.6 D1

Answer: The nearest whole number is 3, but it will be between 3 & 4.

9 < 12 < 1632 = 9 42 = 16

9 + 3 = 12 and 16 - 12 = 4 9 is closer (by 1)

≈ 3122 b) Estimate to two decimal places.12

Use a calculator: 3.46

3) Evaluate without a calculator 4 10 10

8.6 D1

Answer: 40

4 10 10

10 10 10 Remember: 4 · 10

4) Solve x2 = 81

2 81x x = ± 9

10ft

a) 7cm b) 7ft

5cm

2 2 210 7 c 2 2 2b 7 5

5) Solve for the missing side to the nearest tenth.8.6 D1

52 + b2 = 72

b2 = 72 - 52

b ≈ 4.9cm

102 + 72 = c2

12.2ft ≈ c

149 = c

24b

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: 133x x

8.6 D2

3 33 3 3x x x x x

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 200s

8.6 D2

3 · 3 · 3 = 27 so the cube root of 27 = 3

On a calculator type 81^(1/3) ≈ 4.3

s3 = 200 s ≈ 5.85Check: 5.853 ≈ 200

24) 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:

8.6

= 8 exact answer2

2

≈ 11.31 approximate answer

5125

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

More with Square Roots

7) =

8) =

9) 5(2 + 3 ) =

5 9 2 16

4 9 2 16

5·3 + 2·4 = 23

(36).5 + (32).5 = 6 + ≈ 11.66 32

5 20 5(2) + 5(3) = 10 +15 ≈ 89.44 5 20 5 20

8.6

8.4 – 8.6 ReviewSimplify

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)

2 6

4 3 4

3

12

x z y

x y z 2 24

z

x y

12

1

k

16y4

-1024x15y5z20

32

3

x

3

27

8x

positive

15 3.87

y y

24

Rev 2

8.4 – 8.6 ReviewWrite an exact value and a value approximated to the

nearest hundredth.

9) =

10) =

13) solve for x

11)

12) =

1024 24 01000 10 1 ≈ 31.621 1

2 23 34

7 7

12 1.71

7

3 125 = 251

3 2

3 4 2

2

5

xy

x y z

5

2

x y

z

2 27

3 8

x

(hint: use a negative exponent!)

x = -3

Rev 2

8-7 Multiplying and Dividing Square Roots

Objectives: Simplify square roots.

Property:

Products and Quotients of Square Roots: for all nonnegative real numbers a, b and c,

a b a b

8.7

c c

aaThe square root of

any number that isnot a perfect squareis an irrational number.

1) Verify that

a. by finding decimal approximations.

b. by squaring each side.

50 10 5

50 7.07

10 3.16, 5 2.24 AND 3.16 2.24 7.08

50 10 5

2250 10 5

2 250 10 5

50 = 10 · 5 = 50√

8.7

Property of Equality

(ab)2 = a2 · b2 AND 2x x

2) Simplify 150

8.7

150 25 6

150 25 6 25 6

5 6Step 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

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

8.7

6

4

a2 + b2 = c2 62 + 42 = c2

52 = c2

52

4 132 13

= c

= c

= c

Pythagorean Theorem

Substitution

Simplify the right side

Property of Equality

Simplify the radical

4) Simplify 8.7

4 16 3

8

4 4 3

8

1 3

2

4( 1 3)

4 2

4 48

8

Find factors of the radical

Simplify the radical

UNdistribute 4 from each termand cancel.

5) Assume a and b are positive. Find and simplify the result.12 3a b

8.7

12 3a b

36ab

6 ab

4 3 3a b

2 3 3a b

2 3 3 ab

Method 1 Method 2

22 3 ab

2 3 ab

6) Assume a and b are positive. Simplify

3 632a b

8.7

2 2 2 216 2aa b b b34 2ab a

7) Given n ≥ 0, simplify 211n 11n

8) Simplify4

8

3

x

x2

2 2 3

3 3

x

x

2

2 6

3

x

x

2 22 26 21 b ba ba

4 2aabbb

9) Solve y2 + 9 = 33 and simplify the answer.y2 = 24

y = 24

y = 2 6

10) Simplify

11)

2 10 3

6 5

60

30 60

30 2

27

10

9

16

y

z

13

5

3

4

y y

z

8.7

8-8 Distance in a PlaneObjectives: 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 = 2 22 1 2 1( ) ( )x x y y

8.8

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

2 22640 1056 ≈ 2843.37 ft

2304 3509 0948 35

NY@tlas Stephan Van Dam

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.

180

6 5Answer:

2 2( 6 6) ( 2 4)

36 5

Formula

2 22 1 2 1( ) ( )d x x y y

Substitution

Simplify what’s under the radical

Find the perfect square factor

Simplify the radical

8.8

L N

M

Distance = x2 –x1

Distance = y2 –y1

c2 = a2 + b2 c = 2 2a b

Pythagorean Theorem

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.

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

2 2(4 7) (2 11)

8.8

5 2Answer:

90 3 10 ≈ 9.49=

2 2( 5 2) (4 ( 2)) 85 ≈ 9.22

Substitution2 2( 11 ( 6)) (2 7)

50 25 2 Simplify and find perfect square…

Write the Formula2 22 1 2 1( ) ( )x x y y

Write the Formula2 22 1 2 1( ) ( )x x y y

=

=

Write the Formula2 22 1 2 1( ) ( )x x y y

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.

2 2( 5 6) (1 ( 2))

8.8

121 9

130≈ 11.4 miles

Think: If camp is (0,0)what are their coordinateswhen they stop?

Extra Practice

6)

7)

8)

25

12

x

3 72

12

13 822.5x y

2

5

12x

2

5 12

12 12x

2

5 2 3

12x

2

5 3

6x

3 36 2

12

3 6 2

12

1 2 2

4

13 890

4

x y

6 43 10

2

x y x

8.8

8-9 Remembering Properties of Exponents and Powers

Objectives: Evaluate integer powers of real numbers.

Test a special case to determine whether a pattern is true.

8.9

1) Write as a simple fraction.2

7

8

64 151

49 49

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

8.9

3) Use a counter example to show that

≠ + aa + b b

25 ≠ 3 + 4

9 169 + 16

Let a = 9 and b = 16

≠

5 ≠ 7

8.9

4) Simplify

5) Tell whether the pattern is true.

6) Simplify to an exact value .

2 25 3

3 5

d e

e d

2 2(2 )(4 ) 2 2x y xy

45

3

d

e

2 24 2x y

2 2xy

4 8 53 112m n p

2 4 212 7m n p p

8.94

4

625

81

d

e

7) Simplify

8) Simply

9) Simplify

99

11

12

92 3

3

10

5 2

85

5

xy

x y

6

2

17y

x

= 3

8.9

Chapter Review

1) Simplify

a. Product of Powers Property: bm ∙bn = bm+n

x2 ∙ x-2 =

(y2 y4 )y3

= 1

= y9

Chpt Rev

b. Power of a Power Property: (bm)n = bmn

Power of a Product Property: (ab)n = an bn

(2x3 y2)4

c. Negative Exponent Property: b-n =

4-3 =

2

1

3

31

5

1nb

= 24x12y8 = 16x12y8

3

1

4 = 53 = 32

Chpt Rev

d. Quotient of a Powers Property:

6

2

7

7

7

3

12

8

h

h

e. Power of a quotient Property:n n

n

a a

b b

2

36

x

= 74

43

2

h

2

12

x

Chpt Rev

Simplify the exponents and negatives

2) · =

3) (M4)-3 =

4) (-32n-5)2 =

5) -(-50(p-2)2)-3 =

5

1

7

8.4

3

1

7 8

1

7

M-12 = 1/M12

34n-10 = 34 / n10

p12

Chpt Rev

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.

a) How many different forms of the phrase are possible?

b) Write your answer in scientific notation.

c) What is the probability that the fans spelled the phrase correctly when they first sat down?

7! = 7·6·5·4·3·2·1 = 5040

5040 = 5.04 x 103

1

5040

Chpt Rev

7) Simplify

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

8) Find a counterexample.

Define it… ex: (x+y)2 ≠ x2 + y2

-343a3 -7a3 49a2 -49a

Let x = 3 & y = 4(3+4)2 ≠ 32 +44 49 ≠ 25

Chpt Rev

9) Evaluate an expression.

10) Rewrite without negative exponents.

4r-3 s2 t-1

2 7 3 0 4

4 3 2 3 2

3 6

18

xy z x y z

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