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Rational Numbers

Chapter 1, Lesson 1

Vocabulary

Complete this graphic organizer. Rational Number

Define in your own words.

Percent Fraction

Decimal Mixed Number

Rational Numbers

All rational numbers are written as a RATIO.

Example 1.

During a recent regular season, a Texas Ranger

baseball player had 126 hits and was at bat 399

times. Write a fraction in the simplest form to

represent the ratio of the number of hits to the

number of at bats. 126

399 =

6

19

Rational Numbers

Repeating vs. Terminating Decimals

Rational Numbers

Repeating Decimal

Terminating Decimal

1

2 0.50000… 0.5

2

5 0.40000…. 0.4

5

6

0.8333… Does not terminate

(0.83)

Example 1

Write each fraction as a mixed number as a decimal.

a. 5

8

5 ÷ 8 = 0.625

b. -12

3

;5

3 = -5 ÷ 3 = -1.6

Got it? 1

Write each fraction as a decimal.

1. 3

4 2. -

2

9

3. 4 13

25 4. 3

1

11

Example 2

In a recent season, St. Louis Cardinals first baseman

Albert Pujols had 175 hits in 530 at bats. To the

nearest thousand, find his batting average.

We need to the number of hits, 175, by the number

of at bats, 530.

175 ÷ 530 = 0.3301886792

Round to the nearest thousand.

0.330

Got it?

In a recent season, NASCAR driver Jimmie Johnson

won 6 of the 36 total races held. To the nearest

thousandth, find the part of races he won.

Example 3

Write 0.45 as a fraction.

0.45 = 45

100 =

9

20

Example 4

Write 0.5 as a fraction.

N = 0.55555…

Multiply each side by 10.

10N = 10(0.555555….)

10N = 5.5555…..

- N = 0.55555….

Subtract N = 0.55555… to eliminate the repeating part.

9N = 5

N = 5

9

Example 5

Write 2.18 as a mixed number.

N = 2.18181818…

Multiply each side by 100.

100N = 100(2.1818181818….)

100N = 218.181818…..

- N = 2.181818….

Subtract N = 0.181818… to eliminate the repeating part.

99N = 216

N = 216

99 = 2

2

11

Homework

Independent Practice: 1 – 10, 14 – 15, 17, 19

Powers and Exponents

Lesson 2

Saving money Yogi decided to start saving money by putting a penny in his

piggy bank, then doubling the amount he saves each week.

1. Complete the table.

2. How many 2’s are multiplied to find his savings in Week

4? Week 5?

3. How much will he save in Week 8?

Week 0 1 2 3 4 5

Weekly Savings $0.01 $0.02

Total Savings $0.01 $0.03

$0.04

$0.07

$0.08

$0.15

$0.16

$0.31

$0.32

$0.63

Write and Evaluate Powers

2 2 2 2 = 24

4 factors

Read and Write Powers

Power Words Factors

31 3 to the first power 3

32 3 to the second power 3 3

33 3 to the third power 3 3 3

3n 3 to the nth power

3 3 3 3 …3

n factors

Example 1

Write each expression using exponents.

a. (-2) (-2) (-2) 3 3 3 3

There are three (-2)’s and four 3’s.

(-2)3 34

b. a a b b a

There are three a’s and two b’s.

a3 b2

Example 2

Evaluate. (2

3)4

2

3

2

3

2

3

2

3 =

2 𝑥 2 𝑥 2 𝑥 2

3 𝑥 3 𝑥 3 𝑥 3

= 16

81

Got it?

(1

5)3 =

1

125

Example 3

The deck of a skateboard has an area of about 25 7

square inches. What is the area of the skateboard

deck?

25 7

2 2 2 2 2 7

32 7

224 square inches

Example 4

Evaluate each expression if a = 3 and b = 5.

a. a2 + b4

32 + 54

9 + 625

= 634

b. (a – b)2

(3 – 5)2

(-2)2

(-2)(-2) = 4

Multiply and

Divide Monomials

Lesson 3

Monomials

Monomial: a number, variable, or a product of a

number and variable

Examples:

32 74 a4b8 3x2y g

Law of Exponents

c c c c c = c5

c5 c4 = (c c c c c) (c c c c)

= c9

What did you do to the exponents?

ADD THE EXPONENTS

Product of Powers

Words: To multiply powers with the same base,

add their exponents.

Examples:

24 23 = 24+3 or 27

am an = am+n

Example 1 - Simplify

a. c3 c5

c3 c5

c3 + 5 = c8

b. -3x2 4x5

-3x2 4x5

(-3)(4) x2 x5

-12x7

Law of Exponents

r 4 = r r r r

r2 = r r

= r2

What did you do with the exponents?

SUBTRACT THE EXPONENTS

quotient of Powers

Words: To divide powers with the same base,

subtract their exponents.

Examples:

37

33 = 37 – 3 = 34

𝑎𝑚

𝑎𝑛 = am – n


a.48

42

48

42 = 46 = 4,096

b.12𝑥5

2𝑥3

12𝑥5

2𝑥3 = 6x2

Powers of Monomials

Lesson 4

Power of a Power

Words: To find the power of a power, multiply

the exponents.

Examples:

(52)3 = 52 x 3 = 56 (am)n = am n

(64)5 = (64)(64)(64)(64)(64)

5 factors

Example 1

Simplify.

a. (84)3

84 x 3

812

b. (k7)5

k7 x 5

k35

Power of a Product

Words: To find the power of a product, find the

power of each factor and multiply.

Examples:

(6x2)3 = 63 (x2)3 = 216x6

(6x2)3 = (6x2)(6x2)(6x2)

3 factors

Example 2

Simplify.

a. (4p3)4

44 p3x4

256p12

b. (-2m7n6)5

(-2)5 m75 n65

-32m35n30

Negative Exponents

Lesson 5

Zero and Negative

Exponents

Words: Any number to the zero power is 1.

Examples:

40 = 1 b0 = 1

Words: Any number to the negative power is the

multiplicative inverse of its nth power.

Examples:

7-3 = 1

73 = 1

343 k-n =

1

𝑘𝑛


a. 6-2

= 1

62 =1

36

b. a-5

= 1

𝑎5

c. 80

= 1

Example 2 Write each fraction using a negative exponent.

a.1

52

= 5-2

b.1

49

= 1

72 = 7-2

Powers of 10

Exponential Form Standard Form How many Zero’s?

103 1,000 3

102 100 2

101 10 1

100 1 0

10-1 𝟏

𝟏𝟎= 𝟎. 𝟏 1

10-2 𝟏

𝟏𝟎𝟎= 𝟎. 𝟎𝟏 2

10-3 𝟏

𝟏𝟎𝟎𝟎= 𝟎. 𝟎𝟎𝟏 3

Example 3

One human hair is about 0.0001 inch in diameter.

Write this decimal as a power of 10.

0.0001 has 4 zeros

0.0001 = 1

10,000=

1

104

= 10-4


53 x 5-5

= 53+(-5)

= 5-2

= 1

52 = 1

25


𝑏2

𝑏6

=b(2 – 6)

=b-4

= 1

𝑏4

Scientific Notation

Lesson 6

Scientific Notation Table

Expression Product

4.7 x 103 = 4.7 x 1000 4,700

4.7 x 102 = 4.7 x 100 470

4.7 x 101 = 4.7 x 10 47

4.7 x 10-1 = 4.7 x 0.1 0.47

4.7 x 10-2 = 4.7 x 0.01 0.047

4.7 x 10-3 = 4 x 0.001 0.0047

Scientific Notation

Words: when a number is written as the product

of a factor and an integer power of 10.

The number must be between 1 and 10.

Symbols: a x 10n, where a is between 1 and 10

Example:

435,000,000 = 4.35 x 108

Two Rules for S.N.

1. If the number is greater than or equal to 1, the

power of 10 is positive.

2. If the number is between 0 and 1, then power

of ten is negative.

Example 1

Write each number in standard form.

a. 5.34 x 104

5.34 x 10,000

move the decimal point 4 times to the right

= 53,400

b. 3.27 x 10-3

3.27 x 0.001

move the decimal point 3 times to the left

0.00327

Example 2

Write each number in scientific notation.

a. 3,725,000

3.725 x 106

b. 0.000316

3.16 x 10-4

Example 3 - comparing

Refer to the table at the right.

Order the countries

according to the amount of

money visitors spent in the

US from greatest to least.

Dollars Spent by International

Visitors in the U.S.

Country Dollars Spent

Canada 1.03 x 107

India 1.83 x 106

Mexico 7.15 x 106

United Kingdom 1.06 x 107 STEP 1:

1.06 x 107 7.15 x 106

1.03 x 107 1.83 x 106 > STEP 2:

1.06 > 1.03 7.15 > 1.83

CORRECT ORDER:

United Kingdom, Canada, Mexico, India

Example 4

If you could walk to the moon at a rate of 2 meters

per second, it would take you 1.92 x 108 seconds to

walk to the moon. Is it more appropriate to report

this time as 1.92 x 108 seconds, or 6.09 years?

Explain.

The measure 6.09 years is more appropriate. The

number 1.92 x 108 seconds is too large of a number

to describe a walk to the moon.

Compute with

Scientific Notation

Lesson 7

Example 1

Evaluate (7.2 x 103)(1.6 x 104). Express in

scientific notation.

Rearrange the numbers (7.2 x 1.6)(103 x 104)

(11.52)(107)

Move the decimal over to that the number is in

scientific notation.

1.152 x 108

Got it?

a. (8.4 x 102)(2.5 x 106)

b. (2.63 x 104)(1.2 x 10-3)

Example 2

In 2010, the world population was about 6,860,000,000.

The population of the United States was about 3 x 108.

About how many times larger is the world population than

the population of the United States?

Estimate 6,860,000,000 ≈ 7 x 109

Find 7 𝑥 109

3 𝑥 108.

(7

3)(101)

2.3 x 10 = 23

The world’s population is about 23 times bigger than the

United States population.

Adding Numbers in Scientific

Notation

a. (6.89 x 104) + (9.24 x 105)

Make each number have the same power of ten.

(9.24 x 105) = 92.4 x 104

Add the numbers. 6.89 + 92.4 = 99.29

= 99.29 x 104

Put this number in scientific notation.

=9.929 x 105

Adding Numbers in Scientific

Notation

b. 593,000 + (7.89 x 106)

Each number must be in scientific notation.

593,000 = 5.93 x 105


(7.89 x 106) = 78.9 x 105

Add the numbers. 78.9 + 5.93 = 84.83

= 84.83 x 105


=8.483 x 106

Subtracting Numbers in

Scientific Notation

(7.83 x 108) – 11,610,000

Each number must be in scientific notation.

11,610,000 = 1.161 x 107


(7.83 x 108) = 78.3 x 107

Subtract the numbers. 78.3 - 1.161 = 77.139

= 77.139 x 107


=7.7139 x 108

Got it?

a. (8.41 x 103) + (9.71 x 104)

b. (1.263 x 109) - (1.525 x 107)

c. (6.3 x 105) + 2,700,000

Roots

Lesson 8

Vocabulary

Square Root:

A number is one of its two equal factors.

121 = 11

Perfect Square:

Squares of integers: 1, 4, 9, 16, 25, 36, 49, 64…

Also,

(-1)2 = 1 (-2)2 = 4 (-6)2 = 36

Example 1 A. 64 = 8

B. ± 1.21 = ± 1.1 or you could say 1.1, -1.1

C. -25

36 = -

5

6

D. −16 there are not real square roots

Got it? 1

A.9

16

B. ± 0.81

C. - 49

D. −100

Example 2

Solve t2 = 169. Check your solutions.

t2 = 169

𝑡2 = ± 169

t = ± 13

132 = 169 (-13)2 = 169

It checks!!

Got it? 2

A. 289 = a2

B. m2 = 0.09

C. y2 = 4

25

Cube Roots

Cube root: a number is one of three equal factors.

8 = 2 • 2 • 2 = 23

83

= 2

Perfect Cubes: 1, 8, 27, 64, 125, 216…

Also,

-1, -8, -27, -64, -125, -216…

Example 3

A. 1253

= 5

5 • 5 • 5 =125

B. −273

= -3

-3 • -3 • -3 = -27

Got it? 3

A. 7293

B. −643

C. 10003

Example 4

Dylan has a planter in the shape of a cube that holds

8 cubic feet of potting soil. Solve the equation 8 = s3

to find the side length s of the container.

8 = s3

83

= 𝑠33

2 = s

Got it? 4

An aquarium in the shape of a cube that will hold 25

gallons of water has a volume of 3.375 cubic feet.

Solve s3 = 3.375 to find the length of one side of the

aquarium.

Estimating Roots

Lesson 9

Estimating Square and

cube Roots

Non-perfect squares can be estimated.

8 is between 4 and 9

8 is closest to 3 since 8 is closest to 9.

Let’s make a number line.

Example 1

Estimate 83 to the nearest integer.

81 < 83 < 100

83 is closest to 9.

Got it? 1

A. 35

B. 170

C. 44.8

Example 2

Estimate 3203

to the nearest integer.

Find all the cube roots

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

Where does 320 fall?

216 < 320 < 343

3203

is closest to 7

Got it? 2

A. 623

B. 253

C. 129.63

Example 3

Wyatt wants to fence in a square portion of the yard

to make a play area for his new puppy. The area

covered is 2 square meters. Approximately, how

much fencing should Wyatt buy?

Each side is 2, so the perimeter is 4 2.

4 2 is closest to 6 yards.

Got it? 3

Sue wants to fence in a square portion of the yard to

make a play area for her new puppy. The area

covered is 3 square meters. Approximately, how

much fencing should Sue buy?

Example 4

The golden rectangle is found frequently in the

nautilus shell. The length of the longer side divided

by the length of the shorter side is equal to 1: 5

2.

Estimate this value.

5 is closest to 2. So, 1:2

2 equals 1.5.

1: 5

2 estimates to 1.5.

“Little Subset” Verse: Give me a number that’s rational Like any fraction that hurts Accepting positive or negative Are you ready…for two thirds? Or I’ll take the terminating decimal .15, it will be If it’s repeating, it’s sensible So How about, .333333333 Chorus: Hey little subset, I’m a real number The big super-set, rational and irrational Hey smaller subset You call this place an integer? It’s bigger than the whole numbers and counting without the zeros

A rational subset are integers They walk this number line Go both directions from zero They go left, they go right Now, take the positive integers And let’s give them a name zero, 1,2,3,4,5 etc… That’s the whole number game (Chorus) Bridge: Bummed irrational numbers Feel such heavy shame They’re real, but that’s just not the same They envy subsets that complain So they complain blah blah blah blah blah Verse 3: We can’t be written as fractions Else we’d be rational We don’t repeat and/or terminate Like Pi, 3.14159265… (Chorus)

Compare Real

Numbers

Lesson 10

Real Numbers

Example 1

Name all sets of numbers in which each real

number belongs.

A. 0.252525… it has a pattern so it’s rational

B. 36 it equals 6, so it’s a natural, whole,

integer and rational

C. - 7 does not repeat, so it is irrational

Got it? 1

A. 10

B. -2 2

5

C. 100

Example 2

Fill in the blank with <, >, and = to make a true

statement.

A. 7 22

3

B. 15.7% 0.02

<

>

Got it? 2

Fill in the blank with <, >, and = to make a true

statement.

A. 11 31

3

B. 17 4.03

C. 6.25 250%

Example 3

Order the set { 30, 6, 5 4

5, 5.3666…} from least to

greatest.

30 ≈ 5.48

6 = 6

5 4

5 = 5.8

5.3666… ≈ 5.36

5.366…, 30, 5 4

5, and 6

Got it? 3

Order -7, - 60, -7.7, and - 66

9 from least to greatest.

Example 4

On a clear day, the number of miles a person can see

to the horizon is about 1.23 times the square root of

his or her distance from the ground in feet. Suppose

Frida is at the Empire Building observation deck at

1,250 feet and Kia is at the Freedom Tower

observation deck at 1,362 feet. How much farther

can Kia see then Frida?

Frida: 1.23 x 1,250 ≈ 43.05

Kia: 1.23 x 1,362 ≈ 45.51

45.51 – 43.05 = 2.46 miles

vocabulary

base

cube root

exponent

irrational number

monomial

perfect cube

perfect square

power

radical sign

real number

rational number

repeating decimal

scientific notation

square root

terminating decimal

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