featurelesson course 3 lesson main find three different odd numbers whose sum is 21. list all...

FeatureLesson

Course 3Course 3

LessonMain

Find three different odd numbers whose sum is 21. List all possibilities.

1, 3, 17; 1, 5, 15; 1, 7, 13; 1, 9, 11; 3, 5, 13; 3, 7, 11; 5, 7, 9

LESSON 2-1LESSON 2-1

FactorsFactors

Problem of the Day

2-1

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LessonMain


(For help, go to Lesson 1-4.)

Simplify each expression.

2. –10(10) 3. –8(– 7)

4. 5(– 4)(– 2) 5. –1 • 1 • 0

FactorsFactors

1. Vocabulary Review When you multiply two numbers, the result is called the ? .

Check Skills You’ll Need


2-1

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Solutions

1. product 2. –100 3. 56 4. 40 5. 0


FactorsFactors


2-1

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Identify each number as prime or composite. Explain.

FactorsFactorsLESSON 2-1LESSON 2-1

a. 57 Composite; the sum of the digits is 12, which is divisible by 3.

b. 1,354 Composite; the number is divisible by 2.

c. 43 Prime; the number is divisible only by 43 and 1.

d. 975 Composite; the number is divisible by 5.

Quick Check

Additional Examples

2-1

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Use a factor tree to find the prime factorization of 588.


The prime factorization of 588 is 2 • 2 • 3 • 7 • 7, or 22 • 3 • 72.

The number 588 is divisible by 2 because the units digit is 8.

588

1472prime

493prime

Stop when all factors are prime.

7 7prime

2942prime

Begin the factor tree with 2 • 294.

Quick Check

Additional Examples

2-1

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LessonMain

Find the GCF of 55 and 231.


Step 1 Find the prime factorization of each number.

Step 2 Find the product of the common prime factors of each number.

55 = 5 • 11

231 = 3 • 7 • 11

The only common prime factor is 11. The GCF of 55 and 231 is 11.

55

115

231

773

117

Quick Check

Additional Examples

2-1

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36: 1, 2, 3, 4, 6, 9, 12, 18, 36

32: 1, 2, 4, 8, 16, 32

The factors 1, 2, and 4 are common to both numbers.

36: 1, 2, 3, 4, 6, 9, 12, 18, 36

32: 1, 2, 4, 8, 16, 32

The GCF is 4.

36: 1, 2, 3, 4, 6, 9, 12, 18, 36

32: 1, 2, 4, 8, 16, 32

Begin by finding the factors of 36 and 32.

A band with 36 members is marching with a 32-member

band. If the two bands are to have the same number of columns,

what is the greatest number of columns in which you could

arrange the two bands?


So, 4 is the greatest number of columns in which you can arrange the bands.

Quick Check

Additional Examples

2-1

FeatureLesson

Course 3Course 3

LessonMain

FactorsFactors

Write the prime factorization of each number.

1. 24 2. 27 3. 31

Find the GCF of each pair of numbers.

4. 4 and 14 5. 18 and 27

23 • 3


33 31

2 9

Lesson Quiz

2-1

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At the Red Valley Sports Camp, 15 kids went horseback riding, 14 played tennis, 23 went hiking, and the rest of the campers stayed in their cabins. If 83 kids were in the camp, how many stayed indoors?

31


Equivalent Forms of Rational NumbersEquivalent Forms of Rational Numbers

Problem of the Day

2-2

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Find the GCF of each pair of numbers.

2. 6, 12 3. 8, 12 4. 25, 50 5. 36, 40

1. Vocabulary Review Name the prime factorization of 100.



2-2

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Equivalent Forms of Rational NumbersEquivalent Forms of Rational NumbersLESSON 2-2LESSON 2-2

Solutions

1. 2 • 2 • 5 • 5

2. 6 = 2 • 3, 12 = 2 • 2 • 3; GCF = 6

3. 8 = 2 • 2 • 2, 12 = 2 • 2 • 3; GCF = 2 • 2 = 4

4. 25 = 5 • 5, 50 = 2 • 5 • 5; GCF = 5 • 5 = 25

5. 36 = 2 • 2 • 3 • 3, 40 = 2 • 2 • 5; GCF = 2 • 2 = 4


2-2

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Write in simplest form using the GCF.


138150

Divide the numerator and the denominator by the GCF.

138150

138 ÷ 6150 ÷ 6=

Simplify. The numbers 23 and 25 are relatively prime.

=2325

The GCF of 138 and 150 is 6.

Quick Check

Additional Examples

2-2

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Write in simplest form using prime factorization.


60126

Divide the common factors. 2 • 2 • 3 • 52 • 3 • 3 • 7

1 1

1 1

=

Simplify.1021=

Write the prime factorizations of the numerator and denominator.

60126

2 • 2 • 3 • 52 • 3 • 3 • 7=

Quick Check

Additional Examples

2-2

FeatureLesson

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Write each batting average as a decimal.


a. Joe made 4 hits in 20 times at bat.

Joe’s batting average was .200.

b. Pat made 6 hits in 33 times at bat.

Pat’s batting average was about .182.

Write the batting average as a fraction.4

20

Divide the numerator by the denominator. This is a terminating decimal.

0.2

Write the batting average as a fraction.6

33

Use a calculator. This is a repeating decimal.0.18181818

Quick Check

Additional Examples

2-2

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Write 3.225 as a mixed number.


3.225 = Write as a fraction with the denominator 1. 3.225

1

Since there are 3 digits to the right of the decimal, multiply the numerator and denominator by 103 or 1,000.

=3,2251,000

Simplify using the GCF, 25.=3,225 ÷ 251,000 ÷ 25 =

12940

Write as a mixed number.= 39

40

Quick Check

Additional Examples

2-2

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LessonMain


Write each as a fraction in simplest form.

1. 2. –

3. Write as a decimal.

Write each decimal as a mixed number or fraction insimplest form.

4. 2.75 5. 0.4

57


3042

23

–

34

2

1218

216

0.125

25

Lesson Quiz

2-2

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Express 4 days, 12 hours in minutes.

6,480 min


Comparing and Ordering Rational NumbersComparing and Ordering Rational Numbers

Problem of the Day

2-3

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Use the GCF to write each fraction in simplest form.

2. 3. 4. 5.1220

1555

1664

50550

1. Vocabulary Review Explain what the numerator of a fraction represents.



2-3

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Solutions

1. The numerator represents a part of the whole.

2. 3.

4. 5.

Comparing and Ordering Rational NumbersComparing and Ordering Rational NumbersLESSON 2-3LESSON 2-3

12 ÷ 420 ÷ 4

= 35

16 ÷ 1664 ÷ 16

= 14

15 ÷ 555 ÷ 5

= 311

50 ÷ 50550 ÷ 50

= 111


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Which is greater, or ?


718

512

Since the LCM of 18 and 12 is 36, the LCD of the fractions is 36.

Multiples of 18: 18, 36

Multiples of 12: 12, 24, 36

List multiples of each denominator to find their LCD.

Multiply the numerator and denominator by 2.7

187 • 2

18 • 2=

Simplify.1436=

Additional Examples

2-3

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(continued)


Multiply the numerator and denominator by 3.5

125 • 3

12 • 3=

Simplify.1536=

Since1536

1436> , >

718 .

512

Quick Check

Additional Examples

2-3

FeatureLesson

Course 3Course 3

LessonMain

The Eagles won 7 out of 11 games while the Seals won 8

out of 12 games. Which team has the better record?


Change each fraction to a decimal. Compare the two decimals.

Since 0.666 > 0.636, the Seals have the better record.

Divide. Use a calculator.Eagles:

711 0.636363

Seals:8

12 0.666667

Quick Check

Additional Examples

2-3

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Order –0.175, , – , 1.7, –0.95 from least to greatest.


23

58

Then graph each decimal on a number line.

The order of the points from left to right gives the order of the numbers from least to greatest.

–0.95 < –0.625 < –0.175 < 0.667 < 1.7

So, –0.95 < – < –0.175 < < 1.7.23

58

Write each fraction as a decimal.

23 0.667

58 = –0.625–

Quick Check

Additional Examples

2-3

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Compare. Use <, >, or =.

1. 2.

3. 4. 0.35

5. Order 0.17, , –0.3, 0, and – from least to greatest.

6. A survey found that 75 out of 125 men and 88 out of 136 women prefer comedy films over action films. Which group prefers comedy over actions films more?

< >


= =

–0.3, – , 0, 0.17, 14

15

5 12

8 15

45

8 11

1550

36 120

7 20

15

14

women

Lesson Quiz

2-3

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Four divers competed in the belly-flop contest. The bigger the splash the better they do. John made a bigger splash than Bo. Allison came in third. Jennifer came in first with the biggest splash. In what order did the divers finish?

Jennifer, John, Allison, Bo


Adding and Subtracting Rational NumbersAdding and Subtracting Rational Numbers

Problem of the Day

2-4

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1. Vocabulary Review Which numbers are integers: 2, 4.5, 0, –6, ?


2. –9 – 1 3. 10 – 100 4. 12 + (– 2)


13



2-4

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Solutions

1. 2, 0, –6 2. – 10 3. – 90 4. 10




2-4

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A recipe calls for cup white flour and cup wheat flour.

How many total cups of flour are used?

Adding and Subtracting Rational NumbersAdding and Subtracting Rational NumbersLESSON 2-4LESSON 2-4

13

34

You need , or 1 , cups of flour.1312

112

13

1 • 43 • 4+

34 = +

3 • 34 • 3

Write equivalent fractions with the same denominator.

412= +

912 Simplify.

=1312 Add the numerators.

Quick Check

Additional Examples

2-4

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Find – .


34

910

The LCM of 10 and 4 is 20, so the LCD of and is 20.9

1034

910 Write equivalent fractions using

the LCD.

34– =

1820

1520–

Subtract the numerators.18 – 15

203

20==

Quick Check

Additional Examples

2-4

FeatureLesson

Course 3Course 3

LessonMain

Find 6 + 8 .


34

23

Method 1 Use improper fractions.

Write each mixed number as an improper fraction.

6 + 8 = +274

34

23

263

Write equivalent fractions using the LCD, 12.= +8112

10412

Add the numerators.=18512

Change the improper fraction to a mixed number.

= 155

12

Additional Examples

2-4

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(continued)


Method 2 Rewrite the mixed numbers using common denominators.

912

34

236 + 8 = 6 + 8

812

Rewrite each mixed number using the LCD, 12.

1712= 14 + Add the integers and the fractions.

512= 14 + 1

Change the improper fraction to a mixed number.

Add the integers.5

12= 15

Quick Check

Additional Examples

2-4

FeatureLesson

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LessonMain

On a 50-foot roll of cable, 15 ft are left. How many feet of

cable were used?


34

Let t = the amount used.

Words amount left + amount used = original amount

34Equation 15 + t = 50

Additional Examples

2-4

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(continued)


Subtract 15 from each side.t = 50 – 1534

34

The amount of cable used was 34 ft.14

15 + t = 5034

Rewrite 50 as 49 + , or 1. Subtract. t = 49 – 15 = 3444

34

44

14

Quick Check

Additional Examples

2-4

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Find each sum or difference.

1. – 2. + 3. – 4. 4

5. It snowed 2 in. on top of 4 in. of snow already on the ground.

How deep is the snow now?

13

7 in.


89

59

45

23

7838

7 15

1 7 12

3

12

56

12

12

– 114

Lesson Quiz

2-4

FeatureLesson

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Express 106,457,086,299 in words.

one hundred six billion, four hundred fifty-seven million, eighty-six thousand, two hundred ninety-nine


Multiplying and Dividing Rational NumbersMultiplying and Dividing Rational Numbers

Problem of the Day

2-5

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2. 3. 4. 5.1220

7 21

3666

9 81

1. Vocabulary Review A rational number can be written in the form ? , where a and b are integers, and b = 0.



2-5

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Multiplying and Dividing Rational NumbersMultiplying and Dividing Rational NumbersLESSON 2-5LESSON 2-5

Solutions

1. 2. = 3. =

4. = 5. =

7 ÷ 7 21 ÷ 7

13

9 ÷ 9 81 ÷ 9

19

12 ÷ 420 ÷ 4

35

ab

36 ÷ 6 66 ÷ 6

6 11


2-5

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Find • – .


47

512

Multiply the numerators and multiply the denominators.

47

512 • – = –

5 • 412 • 7

Divide the numerator and demoninator by their GCF, 4.

= –5 • 4

12 • 7

1

3

Simplify.= –5

21

Quick Check

Additional Examples

2-5

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LessonMain

Find the product –3 • –2 .


13

56

Estimate: –3 • –2 –3 • (–3) = 913

56

Write as improper fractions.–3 • –2 = – • –13

56

103

176

Divide the numerator and denominator by their GCF, 2.

= 10 • 173 • 6

5

3

Simplify. Write as a mixed number.=859 = 9

49

Check Since 9 is close to 9, the answer is reasonable.49

Quick Check

Additional Examples

2-5

FeatureLesson

Course 3Course 3

LessonMain

One bow takes yards of ribbon. How many bows could

you make from a roll of ribbon that is 12 yards long?


34

12

You can use logical reasoning to solve this problem. You need to find

how many -yd pieces there are in 12 yards.34

12

Divide 12 by .34

12

Write the mixed number as an improper fraction.

25212 ÷ = ÷

34

12

34

Multiply by the reciprocal of .252

43= •

34

Additional Examples

2-5

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(continued)

Divide numerator and denominator by the GCF, 2.

252

43= •

1

2

Multiply. Write the fraction as a mixed number.

= = 16503

23

Since you cannot make of a bow, you can make 16 bows. 23

Quick Check

Additional Examples

2-5

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1. • (– ) 2.

3. ÷ ( ) 4.

5. Solve the equation. 1 r =

6. Megan has 3 quarts of punch. One serving is quart. Does she have enough to serve 15 guests?


29

12

– 58

–2

14

–1

58

45

23

56

12

14

12

No

– 16

– 34

2 • (–1 )13

18

(–1 ) ÷ (1 )78

12

Lesson Quiz

2-5

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LessonMain

Twin primes are pairs of prime numbers who have a difference of 2. For example, 43 – 41 = 2. Name the twin primes between 2 and 35.

5, 3; 5, 7; 11, 13; 17, 19; 29, 31


FormulasFormulas

Problem of the Day

2-6

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LessonMain


FormulasFormulas


Evaluate each expression for w = 2 and t = –3.

2. 4w + t 3. 4(w + t)

4. 4w + 4t 5. – 4t –w

1. Vocabulary Review According to the order of operations, you multiply and divide before you ? and ? .



2-6

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Solutions

1. add; subtract

2. 4(2) + (–3) = 8 + (–3) = 5 3. 4[2 + (–3)] = 4(–1) = –4

4. 4(2) + 4(–3) = 8 + (–12) = –4 5. –4(–3) – 2 =12 – 2 =10

FormulasFormulasLESSON 2-6LESSON 2-6


2-6

FeatureLesson

Course 3Course 3

LessonMain

Find the area of a trapezoid with height of 6 cm and bases

of 5.2 cm and 7.5 cm.


A = h (b1 + b2)Use the formula for the area of a trapezoid.

12

= (6.0) (5.2 + 7.5) Substitute.12

Additional Examples

2-6

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(continued)


The area of the trapezoid is 38.1 cm2.

12= (6.0)(12.7) Add within the parentheses.

= 3(12.7) Multiply from left to right.

= 38.1 Simplify.

Quick Check

Additional Examples

2-6

FeatureLesson

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Find the time it takes a sled-dog team to go 95 miles if their

average rate is 19 mph.


The problem gives distance and rate. Use the distance formula, d = rt where d is the distance traveled, r is the rate of travel, and t is the time spent traveling.

It took the sled-dog team 5 hours to go 95 miles.

d = rt Use the distance formula.

5 = t

Simplify.9519 t=

95 = 19 • t Substitute 95 for d and 19 for r.

Divide each side by 19 to isolate t on the right.9519

19 • t19=

Quick Check

Additional Examples

2-6

FeatureLesson

Course 3Course 3

LessonMain


Which formula can be used to find the diameter d of a circle, given the circumference C?

C Divide each side by to isolate the variable d.

d=

= d Simplify. C

The formula for the diameter of a circle is d = . C

Quick Check

Use the circumference formula for a circle.C = d

Additional Examples

2-6

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FormulasFormulas

27 cm2


126.75 square inches

392.5 miles

1. Find the area of a triangle whose base is 18 cm and height is 3 cm.

2. Amina purchased a circular glass tabletop. The radius of the tabletop is 6.5 inches. Find the area of the tabletop. Use A = r 2 and let = 3.

3. Solve for w in the formula V = wh.

4. Tyrone drove 1570 miles in 4 days. Find the average distance he drove each day.

w = v h

Lesson Quiz

2-6

FeatureLesson

Course 3Course 3

LessonMain

Formulate a set of 5 different numbers whose median is 95 and whose mean is 100.

Sample Answer90, 92, 95, 110, 113


Powers and ExponentsPowers and Exponents

Problem of the Day

2-7

FeatureLesson

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1. Vocabulary Review A ? is an integer that divides another integer with a remainder of 0.

Find the GCF.

2. 12, 16 3. 24, 30 4. 32, 48

5. 120, 144 6. 80, 256




2-7

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Solutions

1. factor 2. 4 3. 6 4. 16 5. 24 6. 16



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Powers and ExponentsPowers and ExponentsLESSON 2-7LESSON 2-7

Write using exponents.

2 • 2 • 2 • 7 • 7

2 is a factor 3 times, and 7 is a factor 2 times. 23 • 72

Quick Check

Additional Examples

2-7

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(–2)6

Multiply. = 64

(–2)6 = (–2)(–2)(–2)(–2)(–2)(–2) The base is –2.

Quick Check

Simplify the expression.

Additional Examples

2-7

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–(2)6

The base is 2. –26 = –(2 • 2 • 2 • 2 • 2 • 2)

Multiply. = –64

Quick Check


Additional Examples

2-7

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38 – (3 • 2)2

38 – (3 • 2)2 = 38 – (6)2 Work inside the grouping symbols.

= 38 – 36 Simplify the power.

= 2 Subtract.

Quick Check


Additional Examples

2-7

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LessonMain


Use the expression to find the radius of a doorway that

has the dimensions s = 3 ft and h = 1 ft.

s2 + h2

2h

s2 + h2

2h32 + 12

2 • 1= Substitute 3 for s and 1 for h.

9 + 12 • 1

= The fraction bar acts as a grouping symbol. Simplify the powers.

102

= Simplify above and below the fraction bar.

Divide.= 5

The radius of the doorway is 5 ft.

Quick Check

Additional Examples

2-7

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1. Write a • a • a • b • b using exponents.

2. Simplify (–4)3. 3. Simplify –25.

4. Simplify (–8 • 5)2 – 92. 5. Evaluate 10 – (5x)2 for x = –2.

6. Find the volume of a child’s wading pool that has a diameter of 6 feet and a height of 1 foot. Use the formula V = r 2h. Use = 3.


a3b2

–64 –32

1,519 –90

27 cubic feet

Lesson Quiz

2-7

FeatureLesson

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LessonMain

Evaluate the following expressions. Write the answers in lowest terms.

a. – = ? b. + = ?


910

110

25

1720

12

34

Scientific NotationScientific Notation

Problem of the Day

2-8

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Scientific NotationScientific Notation

(For help, go to Skills Handbook page 634.)

Multiply.

2. 2 10 3. 4.51 100

4. 1.5 1,000 5. 1.803 10,000

6. 2.39 1,000,000

1. Vocabulary Review An expression using a base and an exponent is a ? .



2-8

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Scientific NotationScientific NotationLESSON 2-8LESSON 2-8

2. 20.0 3. 451.0 4. 1,500.0

5. 18,030.0 6. 2,390,000.0

Solutions

1. power


2-8

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LessonMain

At one point, the distance from Earth to the moon is

1.513431 1010 in. Write this number in standard form.


= 15,134,310,000

At one point, the distance from Earth to the moon is 15,134,310,000 in.

1.513431 1010 = 1.5134310000Move the decimal 10 places to the right. Insert zeros as necessary.

.

Quick Check

Additional Examples

2-8

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LessonMain

The diameter of the planet Jupiter is about 142,800 km.

Write this number in scientific notation.


142,800 = 1 42,800.

The diameter of the planet Jupiter is about 1.428 105 km.

The decimal point moves 5 places to the left..

= 1.428 105 Use 5 as the exponent of 10.

Quick Check

Additional Examples

2-8

FeatureLesson

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LessonMain

Write 4.86 x 10–3 in standard form.


= 0.00486

4.86 x 10–3 = 0.004.86 Move the decimal point 3 places to the left to make4.86 less than 1.

Quick Check

Additional Examples

2-8

FeatureLesson

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LessonMain

Write 0.0000059 using scientific notation.


0.0000059 = 0.000005.9 Move the decimal point 6 places to the right to geta factor greater than 1 butless than 10.

= 5.9 x 10–6 Use 6 as the exponent of 10.

Quick Check

Additional Examples

2-8

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1. Write 7.304 102 in standard form.

2. Write 41,700,000,000 in scientific notation.

3. Write 3.03 x 10–5 in standard form.

4. Write 0.00000127 using scientific notation.

730.4

4.17 1010

0.0000303

1.27 10–6

Lesson Quiz

2-8

featurelesson course 3 lesson main find three different odd numbers whose sum is 21. list all...

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