Chapter 5 Exponents and Polynomials 5.1 Rules for exponents 5.2 Negative Exponents and Scientific Notation 5.3 Adding and Subtracting Polynomials 5.4 Multiplying.

Chapter 5 Exponents and Polynomials

5.1 Rules for exponents5.2 Negative Exponents and Scientific Notation5.3 Adding and Subtracting Polynomials5.4 Multiplying Polynomials5.5 Dividing Polynomials

Simplify each exponential expression. a. 53

b. (–5)2

c. –52

d. (–g)2

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Simplify each expression, using the product rule forexponents.a. r3r4

b. 23 · 25

c. x5x

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Simplify each expression.

a. (4a3b5)(3a7b2)

b. (–5x4y3)(4x2y)

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Simplify each expression, using the quotient rule forexponents.

a.

b.

Back to Table of Contents 5.1-4ab

9

3

a

a

5

3

7

7

Simplify each expression, using the quotient rule forexponents.

c.

Back to Table of Contents 5.1-4c

4x

x

Simplify each expression, using the product or quotientrule for exponents or both.

a.

b.

c.

Back to Table of Contents 5.1-5

4 7

2

a b

ab

2 5 3

2

m n n

mn

5 4 8 5

2 2 7

a b a b

a b a b

Simplify each expression, using the power rule forexponents.

a. (x4)3

b. (23)4

Back to Table of Contents 5.1-6

Simplify each expression.a.

b.

c.

Back to Table of Contents 5.1-7

84

6

a

c

34x

25 47m n

Simplify each expression.a.

b.

Back to Table of Contents 5.1-8ab

3 24 3 32 7x y x y

35 8

3 2

5a b

a b

Simplify each expression.

c.

d.

Back to Table of Contents 5.1-8cd

25

32 3

4

5

m

m n

11

5

7

7

z

z

Rewrite without negative exponents.

a.

b.

Back to Table of Contents 5.2-1ab

3a

26

Rewrite without negative exponents.

c.

d.

Back to Table of Contents 5.2-1cd

5

1

m

49b

Simplify. Write the final answer using positive exponentsonly.

a.

b.

Back to Table of Contents 5.2-2ab

2

8

24

14

a

a

5 415m n

Simplify. Write the final answer using positive exponentsonly.

c.

d.

Back to Table of Contents 5.2-2cd

4 5

2 3

13

7

a b

a b

3 5

4

8

5

x y

x y

Convert the following numbers into scientific notation.a. Microsoft Corporation market capitalization on April 1, 2010: $256,000,000,000.

b. The wavelength of a certain light: 0.0000004 meter.

Back to Table of Contents 5.2-3

Perform the indicated operation. Write the answer in scientific notation.

a.

b.

c.

Back to Table of Contents 5.2-4

84 1.7 10

246 7.0 10

143.4 10

5

The elementary charge, the electric charge carried by a single proton, is approximately 1.602 × 10–19 coulomb. Find the electric charge of 800 protons.

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Perform the indicated operation. Write the answer in scientific notation.a. (7.0 × 108)(2.3 × 1011)

b. (3.6 × 10–9)(2.3 × 10–7)

c.

Back to Table of Contents 5.2-6

15

7

4.8 10

9.5 10

The mass of the sun is estimated at 1.99 × 1030 kg, and the mass of the earth is 5.98 × 1024 kg. Divide the mass of the sun by the mass of the earth to find how much larger the mass of the sun is than that of the earth.

Back to Table of Contents 5.2-7

Identify the terms in each of the following polynomials. Label the polynomial as a monomial, binomial, or trinomial.a. 8x2 + 7x – 10

b. 8a + 28

c. 47m5n3

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For the given polynomials, list the degree of the polynomial. Rewrite the polynomial in descending order.a. x4 + 8x2 – 9

b. 6x – 7 + 2x5 + x3

c.

Back to Table of Contents 5.3-2

120

2a

For the given polynomials, list the degree of each term. Then find the degree of the entire polynomial.a. 11a5b – 5a4b3 + 6b

b. 8m2n2 + mn5 – 9

c. –7

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Classify each of the following as linear equation in one variable, linear equation in two variables, linear expression in one or more variables, or not linear. Explain each classification.a. y = 0.2x + 3.2

b. 3x – 7y + 2

Back to Table of Contents 5.3-4ab

Classify each of the following as linear equation in one variable, linear equation in two variables, linear expression in one or more variables, or not linear. Explain each classification.c. y = 5x2 – 4x + 10

d. 4x – 3

Back to Table of Contents 5.3-4cd

Classify each of the following as linear equation in one variable, linear equation in two variables, linear expression in one or more variables, or not linear. Explain each classification.e. 4xy = 6

Back to Table of Contents 5.3-4e

Add the following polynomials.a. (8a2 + 4a + 5) + (2a2 + 3a + 7)

b. (10m2 – 5mn – 6n2) + (3m2 – 3mn + 2n2)

c. Find the sum of 7x2 + 11 and 12x2 – 8x + 5.

Back to Table of Contents 5.3-5

Subtract the following polynomials.a. (6x2 + 3x + 9) – (4x2 + x – 2)

b. (8a2 – 7a – 9) – (12a2 + 4a – 12)

c. Subtract –5m2 – 8m + 15 from 3m2 + 7m.

Back to Table of Contents 5.3-6

The cost in dollars to assemble b bikes can be estimated by using the polynomial 25b + 100. The cost in dollars for the parts for b bikes can be estimated by using the polynomial 65b + 30.

a. Find a new polynomial that gives the total cost for the parts and assembly of b bikes.

Back to Table of Contents 5.3-7a

The cost in dollars to assemble b bikes can be estimated by using the polynomial 25b + 100. The cost in dollars for the parts for b bikes can be estimated by using the polynomial 65b + 30.

b. Use your new polynomial to find the total cost for parts and assembly of 20 bikes.

Back to Table of Contents 5.3-7b

Write a polynomial that represents the total measurement of the two angles.

Back to Table of Contents 5.3-8

Find each product.a. 5(2x2 + 4x – 9)

b. 7m(6m + 3)

c. 2a(8a2 – 6a + 7)

Back to Table of Contents 5.4-1

Find each product, and simplify.a. (3a – 4)(5a2 + 2a – 8)

b. (x2y + 7xy)(4x5 + 2y – 4)

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Find each product.a. 3x(4x + 5)(2x – 3)

b. –6m(4m + n)(3m – 5n)

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Find each product.a. (a + 2)(a + 7)

b. (2x – 5)(3x + 2)

c. (7m + 4n)(m – 2n)

Back to Table of Contents 5.4-4

The wholesale cost in dollars of c cases of local oranges can be estimated by using the expression 8c + 21. After a large winter storm, the price doubled. Write an expression for the wholesale cost of local oranges after the storm.

Back to Table of Contents 5.4-5

The average amount of milk consumed per person in the United States t years since 2000 can be estimated by using the expression –0.3t + 22.5 gallons per person. The population of the United States in millions can be estimated by using the expression 2.75t + 282 million people.

a. Write a new expression to represent the total amount of milk consumed by the U.S. population.

Back to Table of Contents 5.4-6a

The average amount of milk consumed per person in the United States t years since 2000 can be estimated by using the expression –0.3t + 22.5 gallons per person. The population of the United States in millions can be estimated by using the expression 2.75t + 282 million people.

b. Use the expression found in part a to estimate the amount of milk that will be consumed by the U.S. population in 2015.

Back to Table of Contents 5.4-6b

Write a polynomial that represents the area of the following rectangle.

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Find each product.a. (x + 4)2

b. (3x – 7)2

Back to Table of Contents 5.4-8ab

Find each product.c.

d.

Back to Table of Contents 5.4-8cd

23

5m n

225 3a b

Identify the expression inside the parentheses as a product or a sum. Then multiply each expression. Simplify.a. (7x)2

b. (7 + x)2

Back to Table of Contents 5.4-9

Find each product.a. (a + 6)(a – 6)

b. (2x – 5)(2x + 5)

c. (3p5 + 7)(3p5 – 7)

Back to Table of Contents 5.4-10

Perform the indicated operations, and simplify.a. 8x2 – 9 + (x – 6)(5x + 2)

b. (x – 7)2 – (x + 9)2

Back to Table of Contents 5.4-11

Divide and simplify.

a.

b.

Back to Table of Contents 5.5-1

326 16

8

a a

a

3 2

2

11 21

7

m m

m

Divide and simplify.

a.

b.

Back to Table of Contents 5.5-2

4 3 3 2 2

2 2

14 42 12

7

m n m n m n

m n

220 12 36

4

a a

a

Divide.

a. (x2 + 8x + 15) ÷ (x + 3)

b. (8x2 + 2x – 21) ÷ (2x – 3)

Back to Table of Contents 5.5-3

Find an expression that represents the length of the missing side of this rectangle.

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Divide, and check the result.a. (10x2 – 27x + 30) ÷ (2x – 3)

Back to Table of Contents 5.5-5a

Divide, and check the result.b. (10x3 + 63x2 + 71x – 79) ÷ (x + 3)

Back to Table of Contents 5.5-5b

Divide, and check the result.a. (4x3 – 14 + 21x2) ÷ (x + 5)

Back to Table of Contents 5.5-6a

Divide, and check the result.b. (27x3 – 8) ÷ (3x – 2)

Back to Table of Contents 5.5-6b