Example 1 Use Divisibility Rules

Example 2 Use Divisibility Rules to Solve a Problem

Example 3 Find Factors of a Number

Example 4 Identify Monomials

Determine whether 435 is divisible by 2, 3, 5, 6, or 10.

Number Divisible? Reason

2

3

5

6

10

no The ones digit is 5 and 5 is not divisible by 2.

The ones digit is 5.yes

no The ones digit is not 0.

no 435 is not divisible by 2, so it cannot be divisible by 6.

Answer: So, 435 is divisible by 3 and 5.

yes The sum of the digits is or 12 and 12 is divisible by 3.

Determine whether 786 is divisible by 2, 3, 5, 6, or 10.

Answer: 786 is divisible by 2, 3, and 6.

Student Elections Sonya is running for student council president. She wants to give out campaign flyers with a pen to each student in the school. She can buy “Vote for Sonya” pens in packages of 5, 6, or 10. If there are 306 students in the school and she wants no pens left over, which size packages should she buy?

Size Yes/No Reason

5

6

10

no The ones digit of 306 is not 0 or 5.

yes 306 is divisible by 2 and 3, so it is also divisible by 6. Therefore, there would be no pens left over.The ones digit is not 0. no

Answer: Sonya should buy pens in packages of 6.

Transportation A class of 72 students is taking a field trip. The transportation department can provide vans that seat 5, 6, or 10 students. If the teacher wants all vans to be the same size and no empty seats, what size vans should be used?

Answer: Vans that seat 6 should be used.

List all the factors of 64. Use the divisibility rules to determine whether 64 is divisible by 2, 3, 5, and so on. Then use division to find other factors of 64.

Number 64 Divisible by Number? Factor Pairs

1

2

3

4

5

6

7

8

___no

___no___no___no

yes

yes

yes

yes

Answer: So, the factors of 64 are 1, 2, 4, 8, 16, 32, and 64.

List all the factors of 96.

Answer: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96

Determine whether is a monomial.

Simplify.

Answer: This expression is not a monomial because in its simplest form, it involves two terms that are added.

Distributive Property

Determine whether is a monomial.

Answer: This expression is a monomial because

it is the product of a rational number,

,

and a variable, x.

Determine whether each expression is a monomial.

Answer: monomial

Answer: not a monomial

a.

b.

Example 1 Write Expressions Using Exponents

Example 2 Use Exponents in Expanded Form

Example 3 Evaluate Expressions

Write using exponents.

Answer: The base is 6. It is a factor 4 times, so the exponent is 4.

Write p using exponents.

Answer: The base is p. It is a factor 1 time, so the exponent is 1.

Write (–1)(–1)(–1) using exponents.

Answer: The base is – 1. It is a factor 3 times, so the exponent is 3.

Write using exponents.

Answer: The base is . It is a factor 2 times, so the exponent is 2.

Write each expression using exponents.

Answer: First group the factors with like bases. Then write using exponents.

Answer:

Answer: Answer:

Answer:

Answer:

Write each expression using exponents.

a.

b.

c.

d.

e.

Express 235,016 in expanded form.

Answer:

Step 1 Use place value to write the value of each digit in the number.

Step 2 Write each place value as a power of 10using exponents.

Express 24,706 in expanded form.

Answer:

Answer: 16

4 is a factor two times.

Multiply.

Evaluate .

–2 is a factor 3 times.

Multiply.

Subtract.

Answer:

Replace r with –2.

Evaluate .if

Simplify the expression inside the parentheses.

Evaluate (0)2.

Replace x with 2 and y with –2.

Simplify.

Answer: 0

Evaluate .if and

Evaluate each expression.

Answer: 81

Answer: 84

Answer: –24

a.

b. if

c.

Example 1 Simplify Fractions

Example 2 Simplify Fractions

Example 3 Simplify Fractions in Measurement

Example 4 Simplify Algebraic Fractions

Example 5 Simplify Algebraic Fractions

Write in simplest form.

The GCF of 16 and 24 is or 8.

Answer:

Factor the numerator.

Factor the denominator.

Divide the numerator and denominator by the GCF.

Simplest form

Write in simplest form.

Answer:

Write in simplest form.

Simplify.

Answer:

Divide the numerator and the denominator by the GCF, .

1 1 1 1

1 1 1 1

Write in simplest form.

Answer:

Measurement 250 pounds is what part of 1 ton?

There are 2000 pounds in 1 ton.

Write the fraction in simplest form.

Answer: So, 250 pounds is of a ton.

Simplify.

Divide the numerator and the denominator by the GCF, .

1 1 1 1

1 1 1 1

80 feet is what part of 40 yards?

Answer:

Simplify .

Simplify.

Answer:

1 1Divide the numerator and the denominator by the GCF, .1 1 1

1

Simplify .

Answer:

A B C D

Read the Test Item In simplest form means that the GCF of the numerator and denominator is 1.

Which fraction is written in simplest form?

Multiple-Choice Test Item

Solve the Test Item

Answer: C

Factor.

1 1 1 1

1 1 1 1

A B C D

Answer: D

Which fraction is written in simplest form?

Multiple-Choice Test Item

Example 1 Multiply Powers

Example 2 Multiply Monomials

Example 3 Divide Powers

Example 4 Divide Powers to Solve a Problem

Find .

Check

Answer:

Add the exponents.

The common base is 3.

Find .

Answer:

Find .

Answer:

The common base is y.

Add the exponents.

Find (3p4)(–2p3).

Answer: –6p7

Use the Commutative and Associative Properties.

(3p4)(–2p3) (3 • –2)(p4 • p3)

Add the exponents.–6p7

The common base is p.(–6)(p4+3)

Find each product.

Answer:

a.

b.

Answer:

The common base is 8.

Answer:

Subtract the exponents.

Find .

The common base is x.

Subtract the exponents.

Answer:

Find .

Find

Answer:

Folding Paper If you fold a sheet of paper in half, you have a thickness of 2 sheets. Folding again, you have a thickness of 4 sheets. Continue folding in half and recording the thickness. How many times thicker is a sheet that has been folded 4 times than a sheet that has not been folded?

Write a division expression to compare the thickness.

Subtract the exponents.

Answer: So, the paper is 16 times thicker.

Racing Car A can run at a speed of miles per hour and car B runs at a speed of miles per hour. How many times faster is car A than car B?

Answer: Car A is 2 times faster than car B.

Example 1 Use Positive Exponents

Example 2 Use Negative Exponents

Example 3 Use Exponents to Solve a Problem

Example 4Algebraic Expressions with Negative Exponents

Answer:

Definition of negative exponent

Write using a positive exponent.

Answer:

Write using a positive exponent.

Definition of negative exponent

Write each expression using a positive exponent.

Answer:

a.

b.

Answer:

Write as an expression using a negative exponent.

Answer:

Find the prime factorization of 125.

Definition of exponents

Definition of negative exponent

Write as an expression using a negative exponent.

Answer:

Physics An atom is an incredibly small unit of matter. The smallest atom has a diameter of approximately of a nanometer, or 0.0000000001 meter. Write the decimal as a fraction and as a power of 10.

Answer:

Write the decimal as a fraction.

Definition of negative exponent

Write 0.000001 as a fraction and as a power of 10.

Answer:

Find .

Answer:

Replace r with –4.

Definition of negative exponent

Evaluate .if

Answer:

Evaluate .if

Example 1 Express Numbers in Standard Form

Example 2 Express Numbers in Scientific Notation

Example 3 Use Scientific Notation to Solve a Problem

Example 4 Compare Numbers in Scientific Notation

Express in standard form.

Answer: 43,950

Move the decimal point 4 places to the right.

Answer: 0.00000679

Move the decimal point 6 places to the left.

Express in standard form.

Express each number in standard form.

a.

b.

Answer: 2,614,000

Answer: 0.000803

Express 800,000 in scientific notation.

Answer:

The exponent is positive.

The decimal point moves 5 places.

Express 1,320,000 in scientific notation.

The exponent is positive.

The decimal point moves 6 places.

Answer:

Express 0.0119 in scientific notation.

The exponent is negative.

The decimal point moves 2 places.

Answer:

Express each number in scientific notation.

a. 65,000

b. 3,024,000

c. 0.00042

Answer:

Answer:

Answer:

Space The table shows the planets and their distances from the Sun. Estimate how many times farther Pluto is from the Sun than Mercury is from the Sun.

Planet Distance from the Sun (km)Mercury 5.80 x 107

Venus 1.03 x 108

Earth 1.55 x 108

Mars 2.28 x 108

Jupiter 7.78 x 108

Saturn 1.43 x 109

Uranus 2.87 x 109

Neptune 4.50 x 109

Pluto 5.90 x 109

Explore You know that the distance from the Sun to Pluto is km and the distance from the Sun to Mercury is km.

Plan To find how many times farther Pluto is from the Sun than Mercury is from the Sun, find the ratio of Pluto’s distance to Mercury’s distance. Since you are estimating, round the distanceto and round the distanceto .

Examine Use estimation to check the reasonableness of the results.

Solve Divide

Answer: So, Pluto is about 1.0 102 or 100 times farther from the Sun than Mercury.

Planet Distance from the Sun (km)Mercury 5.80 x 107

Venus 1.03 x 108

Earth 1.55 x 108

Mars 2.28 x 108

Jupiter 7.78 x 108

Saturn 1.43 x 109

Uranus 2.87 x 109

Neptune 4.50 x 109

Pluto 5.90 x 109

Space Use the table to estimate how many times farther Pluto is from the Sun than Earth is from the Sun.

Answer: 30 times farther

Space The diameters of Mercury, Saturn, and Plutoare kilometers, kilometers, and

kilometers, respectively. List the planets inorder of increasing diameter.

First, order the numbers according to their exponents.

Then, order the numbers with the same exponent by comparing the factors.

Answer: So, the order is Pluto, Mercury, and Saturn.

Step 1

Step 2

Mercury and Pluto Saturn

Pluto Mercury

Compare the factors:

Order the numbers , , ,and in decreasing order.

Answer: , , , and.