600 Prerequisite Skills

Multiply and Divide with Scientific NotationMississippi Standard: Multiply and divide numbers written in scientific notation.

You can use scientific notation to simplify computations with very large and/or very small numbers.

To multiply numbers in scientific notation, regroup to multiply the factors and multiply the powers of ten. Then simplify. To multiply the powers of ten, use the

.Product of Powers

Multiplication with Scientific Notation

Evaluate the expression (1.3 � 102)(2.5 � 101).

(1.3 � 102)(2.5 � 101) � (1.3 � 2.5)(102 � 101) Commutative and Associative Properties

� (3.25)(102 � 101) Multiply 1.3 by 2.5.

� 3.25 � 102 � 1 Product of Powers

� 3.25 � 103 Add the exponents.

� 3.25 � 1,000 103 � 1,000

� 3,250 Move the decimal point 3 places.

Evaluate the expression (4.2 � 103)(1.6 � 104).

(4.2 � 103)(1.6 � 104) � (4.2 � 1.6)(103 � 104) Commutative and Associative Properties

� (6.72)(103 � 104) Multiply 4.2 by 1.6.

� 6.72 � 103 � 4 Product of Powers

� 6.72 � 107 Add the exponents.

� 6.72 � 10,000,000 107 � 10,000,000

� 67,200,000 Move the decimal point 7 places.

Product of Powers

Words To multiply powers with the same base, add their exponents.

Symbols Arithmetic Algebra

32 � 35 � 32 � 5 or 37 xa � xb � xa � b

To divide numbers in scientific notation, regroup to divide the factors anddivide the powers of ten. Then simplify. To divide the powers of ten, use the

.Quotient of Powers

Quotient of Powers

Words To divide powers with the same base, subtract their exponents.

Symbols Arithmetic Algebra

�44

8

3� � 48 � 3 or 45 �xx

b

a� � xa � b, x � 0

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Prerequisite Skills 601

ExercisesMultiply or divide. Express using exponents.

1. 51 � 54 2. 65 � 64 3. 102 � 103

4. �22

5

3� 5. �77

6

5� 6. �1100

9

6�

Evaluate each expression. Express the result in scientific notation and standard form.

7. (2.6 � 105)(1.9 � 102) 8. (5.3 � 104)(0.9 � 103)

9. (3.7 � 102)(1.2 � 102) 10. (3.3 � 103)(2.1 � 102)

11. (8.5 � 103)(1.1 � 101) 12. (3.9 � 102)(2.3 � 106)

13. (6.45 � 105)(1.2 � 103) 14. (4.18 � 104)(0.9 � 105)

15. �82.3.77��11003

8� 16. �

86.0.74��11002

5�

17. �91.7.82��11005

9� 18. �

42.6.94��11003

4�

19. �81.3.32��11005

7� 20. �

61..35

��11001

6

0�

21. �14..628��11008

2� 22. �92..04

��11001

8

1�

23. BASEBALL The table shows the 2007 salaries of six Major League Baseball players. About how many times greater is Alex Rodriguez’s salary than Juan Castro’s salary?

24. ASTRONOMY The Sun burns about 4.4 � 106 tons of hydrogen per second. How much hydrogen does the Sun burn in one year? (Hint: one year � 3.16 � 107 seconds)

25. OCEANS The area of the Pacific Ocean is 6.0 � 107 square miles. The area of the Atlantic Ocean is 2.96 � 107 square miles. About how many times greater is the area of the Pacific Ocean than the Atlantic Ocean?

2007 Major League Baseball Salaries

Player

Juan CastroCoco CrispNomar GarciaparraChipper JonesKazuo MatsuiAlex Rodriguez

Source: USA Today

Team

Cincinnati RedsBoston Red SoxLos Angeles DodgersAtlanta BravesColorado RockiesNew York Yankees

Salary(dollars)

9.25 � 105

3.83 � 106

8.52 � 106

1.23 � 107

1.5 � 106

2.27 � 107

Division with Scientific Notation

Evaluate the expression �92.4.15��11003

6�.

�92.4.15��11003

6� � ��

92.4.15

����1100

6

3�� Associative Property

� 4.5 � ��1100

6

3�� Divide 9.45 by 2.1.

� 4.5 � 106 � 3 Quotient of Powers

� 4.5 � 103 Subtract the exponents.

� 4.5 � 1,000 103 � 1,000

� 4,500 Move the decimal point 3 places.

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602 Prerequisite Skills

The Density PropertyMississippi Standard: Develop a logical argument to demonstrate the‘denseness’ of rational numbers.

Examine the number line below. Find another integer that lies between theintegers �2 and 3.

The integers �1, 0, 1, and 2 all lie between �2 and 3 on the number line.On the number line above, find a number that lies between 1 and 2. Their

average, 1�12

�, is one number that lies between 1 and 2.

0 1 2 3 4-2 -1

Find a Number Between Two Given Numbers

Find a number that lies between �13

� and �12

� on the number line below.

One number would be their average.

�12

���13

� � �12

�� � �12

���26

� � �36

�� Rewrite �13

� and �12

� with a common denominator.

� �12

���56

�� Add the numerators.

� �152� Multiply.

The rational number, �152�, lies between �

13

� and �12

�.

Find a number that lies between �7 and �6.5.

One number would be their average.

�12

�[�7 � (�6.5)] � �12

�(�13.5) Add �7 and �6.5.

� �6.75 Multiply.

The rational number, �6.75, lies between �7 and �6.5.

0 156

12

13

16

23

The process above of finding another number between any two given numberscan be continued indefinitely. This suggests the .density property

You can use the density property to solve real-world problems.

Density Property for Rational Numbers

Words Between every pair of distinct rational numbers, there are infinitely many rational numbers.

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Prerequisite Skills 603

ExercisesIdentify a number that lies between points A and B on each number line.

1. 2.

3. 4.

Identify a number that lies between each pair of numbers.

5. 6 and 7 6. �10 and �9 7. �34

� and 1

8. �2 and �1�12

� 9. 4�23

� and 4�34

� 10. �5 and �4�13

�

11. �4 and �3 12. 8.25 and 8.75 13. 15.5 and 16

14. SCHOOL For reading class, Dylan is recording the number of hours he reads

each week. This week, Dylan needs to read between 1�12

� and 2 hours. What

is a possible time between 1�12

� and 2 hours that Dylan can read?

15. CROSS COUNTRY For cross-country practice, the coach told the runners

they needed to run between 5�12

� and 5�34

� miles. Give a possible distance between

5�12

� and 5�34

� miles that a runner can run.

16. Demonstrate the density property for rational numbers with severalexamples of your own.

Apply the Density Property

BAKING Genevieve’s grandmother gave her a family recipe for apple pie.Her grandmother does not use an exact amount of sugar, but told

Genevieve to use somewhere between 1�14

� and 1�12

� cups of sugar. If

Genevieve wants to use an exact amount of sugar that is somewhere

between 1�14

� cups and 1�12

� cups, how much sugar can she use?

One possible amount is their average.

�12

��1�14

� � 1�12

�� � �12

��1�14

� � 1�24

�� Rename �12

� as �24

�.

� �12

��2�34

�� Add the whole numbers and add the fractions.

� �12

���141�� Rewrite 2�

43� as an improper fraction.

� �181� or 1�

38

� Simplify.

So, Genevieve can use 1�38

� cups of sugar.

0 1 2 3 4 5

A B

-4 -3 -2 -1 10

A B

A B

110

210

310

410

510

610

0 0.5 1 1.5 2 2.5

A B

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604 Prerequisite Skills

Identify Properties

Name the property shown by each statement.

(7 � 3x) � 2x � 7 � (3x � 2x)

Associative Property

0 � 5a � 0

Zero Property of Multiplication

Algebraic PropertiesMississippi Standard: Apply algebraic properties in problem-solving.

Review the properties in the table below. These properties can be applied when problem-solving.

Properties

The order in which numbers are added or multiplied does not change thesum or product.

Examples 6 � 7 � 7 � 6 a � b � b � a3 � 8 � 8 � 3 a � b � b � a

The way in which numbers are grouped when added or multiplied does notchange the sum or product.

Examples (2 � 7) � 4 � 2 � (7 � 4) (a � b) � c � a � (b � c)(3 � 4) � 5 � 3 � (4 � 5) (a � b) � c � a � (b � c)

To multiply a sum by a number, multiply each addend by the numberoutside the parentheses.

Examples 2(7 � 4) � 2 � 7 � 2 � 4 a(b � c) � ab � ac(5 � 6)3 � 5 � 3 � 6 � 3 (b � c)a � ba � ca

The sum of any number and 0 is the number.

Examples 7 � 0 � 7 a � 0 � a

The product of any number and 0 is 0.

Examples 9 � 0 � 0 a � 0 � 0

The product of any number and 1 is the number.

Examples 3 � 1 � 3 a � 1 � a

Multiplicative Identity

Zero Property of Multiplication

Additive Identity

Distributive Property

Associative Property

Commutative Property

Use Properties to Simplify Expressions

Simplify each expression. Justify each step.

4 � (x � 13)

4 � (x � 13) � 4 � (13 � x) Commutative Property

� (4 � 13) � x Associative Property

� 17 � x Add 4 and 13.

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Prerequisite Skills 605

6(x � 7)

6(x � 7) � 6(x) � 6(7) Distributive Property

� 6x � 42 Multiply.

ExercisesName the property shown by each statement.

1. 3n � m � m � 3n 2. 0 � 18d � 18d

3. (7y � 8) � 10y � 7y � (8 � 10y) 4. 20xy � 1 � 20xy

5. 3(6a � 7b) � 3 � 6a � 3 � 7b 6. 82 � 0 � 0

Simplify each expression. Justify each step.

7. 1 � (6 � x) 8. 5(6a) 9. 11 � (6 � n)

10. 5(x � 8) 11. 15(4w) 12. 9(x � 2)

13. 9 � 2y � 11 � 5y 14. 4(x � 7) � 2x 15. 11n � 7(2 � 3n)

16. ANIMALS A zebra can run up to 40 miles per hour. An elephant can run up to x miles per hour. Write and simplify an expression to find how many more miles azebra will run in six hours than an elephant.

17. CELL PHONES Seven friends have similar cell phone plans. The price of each plan is $x. Three of the seven friends pay an extra $4 per month for unlimitedtext messaging. Write and simplify an expression that represents the total cost ofthe seven plans.

Apply Properties to Problem Solving

MUSEUMS Three friends are going to the science museum. The cost of admission is $x each. It will cost an additional $4 to view a movie on the 3-D screen. Write and simplify an expression that represents the total cost for the three friends.The cost of admission plus the movie can be represented by (x + 4).

Multiply this cost by the number of friends, 3(x + 4).

3(x � 4) � 3(x) � 3(4) Distributive Property

� 3x � 12 Multiply.

So, the total cost for the three friends is $3x � $12.

MUSEUMS Refer to Example 5. A fourth friend will meet the group of friends at the museum but will not go to the movie. Write and simplify an expression that represents the total cost for the four friends.The cost for the fourth friend is $x. Add this to $3x � $12.

3x � 12 � x � 3x � x � 12 Commutative Property

� 4x � 12 Add.

So, the total cost for the four friends is $4x � $12.

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606 Prerequisite Skills

Make Predictions from Circle Graphs and HistogramsMississippi Standard: Use proportions, estimates, and percentages toconstruct, interpret, and make predictions about a population based onhistograms or circle graph representations of data from a sample.

You can make predictions about a given set of data displayed in a circle graph or histogram. Use percentages to make predictions about data displayed in a circle graph.

Predict from a Circle Graph

The circle graph shows the results of a survey ofthe students in the 8th grade at Oakwood JuniorHigh. If there are 560 students at Oakwood JuniorHigh, how many would you predict to choosereality as their favorite type of television show?

The section of the graph representing students whochose reality is 40% of the circle. So find 40% of 560.

To find 40% of 560, you can use either method.

METHOD 1 Write the percent as a decimal.

40% of 560 � 40% � 560 Write a multiplication expression.

� 0.40 � 560 Write 40% as a decimal.

� 224 Multiply.

METHOD 2 Write the percent as a fraction.

40% of 560 � 40% � 560 Write a multiplication expression.

� �14000

� � �5610

� Write 40% as a fraction. Write 560 as �5160�.

� 224 Multiply.

So, about 224 students at Oakwood Junior High would choose reality as their favorite type of television show.

4%Other

Favorite Type of Television Show

40%Reality

28%Comedy

16%Cartoon

5%Drama

7%Fiction

Predict from a Histogram

The histogram shows the winning times of themen’s 400-meter run in the summer Olympicsfrom 1896 to 2004. Predict the range of speedsthat a runner finishing in first place is mostlikely to be in the next summer Olympics?Explain your reasoning.

The bar at 43.0–44.9 seconds is much higher thanthe others and represents the most winning times.So, the winning speed of the runner in the nextsummer Olympics will most likely be in the 43.0–44.9 second range.

43.0–44.9

45.0–46.9

47.0–48.9

49.0–50.9

51.0–52.9

53.0–54.9

2

4

6

8

10

12

0

Num

ber o

f Win

ners

Time (seconds)

Summer Olympic Men’s 400-Meter RunWinning Times, 1896–2004

Source: The World Almanac

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Prerequisite Skills 607

ExercisesCARS For Exercises 1–3, use the circle graph that shows the mostpopular luxury car colors.

1. If a car dealership sold 50 luxury cars in March, predict howmany were white.

2. If a car dealership sold 250 luxury cars in January throughJune, predict how many were black.

3. If a parking garage has 85 luxury cars parked on a given day,predict how many are silver/gray.

VACATION For Exercises 4 and 5, use the circle graph thatshows the results of a survey of the favorite summertimeactivities of 7th grade students at Parson Junior High.

4. If there are 275 students at Parson Junior High, predicthow many would choose visiting an amusement park astheir favorite summertime activity.

5. If there are 150 students at Parson Junior High, predicthow many would choose swimming or going to camp astheir favorite summertime activity.

HISTORY For Exercises 6 and 7, use the histogram that shows the age of U.S. presidents at theirinauguration.

6. Predict the 5-year age range that the next U.S.president will most likely be in at theirinauguration.

7. Predict the 10-year age range that the next U.S.president will most likely be in at theirinauguration.

SCHOOL For Exercises 8 and 9, use the histogram thatshows the test scores of Mrs. Jeng’s first period mathclass. Mrs. Jeng teaches three math classes of thesame level in the morning.

8. Predict the range that students in Mrs. Jeng’ssecond period math class will most likely score.

9. Predict the range that students in Mrs. Jeng’s thirdperiod math class will least likely score.

5%Other

Most PopularLuxury Car Colors

26%Silver/Gray

28%White12%

Black

9%Red

9%Blue

11%LightBrown

55–5

9

50–5

4

45–4

9

40–4

4

2

4

6

8

10

12

14

16

0

Num

ber o

f Pre

side

nts

60–6

4

65–6

9

Age at Inaguration

U.S. Presidents Age at Inauguration

81–9071–8061–7051–60

2

4

6

8

10

12

14

0

Num

ber o

f Stu

dent

s

91–100Score

Mrs. Jeng’s First Period Test Scores

5%Other

Favorite Summertime Activity

32%Amusement

Park

24%Swim

20%Camp

6%Read

13%Beach

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608 Prerequisite Skills

Euler’s FormulaMississippi Standard: Construct three-dimensional figures using manipulativesand generalize the relationships among vertices, faces, and edges (such asEuler’s Formula).

Recall that a three-dimensional figure has a length, width, and depth (or height). Theflat surfaces of a three-dimensional figure are the . The line segments wherethe faces meet are the . The points where the edges intersect are the .

The number of faces, vertices, and edges of a three-dimensional figure arerelated by . Euler’s (OY-luhrz) Formula

verticesedgesfaces

edge

vertex

face

Euler’s Formula

Words In a three-dimensional figure, the sum of the faces F and vertices Vis equal to two more than the number of edges E.

Symbols F � V � E � 2

Verify Euler’s Formula

Determine whether Euler’s Formula is true for the figure below.

The figure has 5 faces, 6 vertices, and 9 edges.

F � V � E � 2 Euler’s Formula

5 � 6 � 9 � 2 Substitute 5 for F, 6 for V, and 9 for E.

11 � 11 ✓ Add. The sentence is true.

Yes, the formula is true for the figure shown.

You can verify Euler’s Formula for three-dimensional figures. For example,the rectangular prism above has 6 faces, 8 vertices, and 12 edges.

F � V � E � 2 Euler’s Formula

6 � 8 � 12 � 2 Substitute 6 for F, 8 for V, and 12 for E.

14 � 14 ✓ Add. The sentence is true.

Since 14 � 14, the formula is true for the rectangular prism.

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Prerequisite Skills 609

ExercisesVerify Euler’s Formula for each figure.

1. 2. 3.

4. 5. 6.

Use Euler’s Formula to find the missing number of faces, vertices, or edges for each three-dimensional figure described.

7. F � ? 8. F � 6 9. F � 7

V � 4 V � ? V � 7

E � 6 E � 12 E � ?

10. F � 2 11. F � 8 12. F � ?

V � 1 V � 18 V � 7

E � ? E � ? E � 11

13. F � 11 14. F � ? 15. F � 10

V � 11 V � 14 V � ?

E � ? E � 21 E � 16

16. DIAMONDS A princess cut diamond has 5 faces and 8 edges. How many vertices are there on a princess cut diamond?

Use Euler’s Formula

A three-dimensional figure has 5 faces and 4 vertices. Use Euler’s Formula to find the number of edges in the figure.

Use Euler’s Formula and solve for E.

F � V � E � 2 Euler’s Formula

5 � 4 � E � 2 Substitute 5 for F and 4 for V.

9 � E � 2 Add 5 and 4.

� 2 � � 2 Subtract 2 from each side.

7 � E Simplify.

The figure has 7 edges.

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