Using the Properties to Simplify Expressions- 123docz.net

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1.8 Using the Properties to Simplify Expressions

The properties of the real numbers can be helpful when we are doing computations. In this section we will see how the properties can be applied in arithmetic and algebra.

U 1 V Using the Properties in Computation

The properties of the real numbers can often be used to simplify computations. For example, to find the product of 26 and 200, we can write

(26)(200)(26)(2 100) (26 2)(100) 52 100 5200.

It is the associative property that allows us to multiply 26 by 2 to get 52, and then multiply 52 by 100 to get 5200.

E X A M P L E 1 Using the properties

Use the appropriate property to aid you in evaluating each expression.

a) 3473565 b) 3 435 1

3 c) 6 284 28

Solution

a) Notice that the sum of 35 and 65 is 100. So apply the associative property as follows:

347(3565)347100 447

1-67 1.8 Using the Properties to Simplify Expressions 67

U 2 V Combining Like Terms

An expression containing a number or the product of a number and one or more variables raised to powers is called a term. For example,

3, 5x, 3x2y, a, and abc

are terms. The number preceding the variables in a term is called the coefficient. In the term 5x, the coefficient of x is 5. In the term 3x2y the coefficient of x2y is 3. In the term a, the coefficient of a is 1 because a1 a. In the term abc the coefficient of abc is 1 because abc 1 abc. If two terms contain the same variables with the same exponents, they are called like terms. For example, 3x2and 5x2are like terms, but 3x2and 5x3are not like terms.

Using the distributive property on an expression involving the sum of like terms allows us to combine the like terms as shown in Example 2.

b) Use the commutative and associative properties to rearrange this product. We can then do the multiplication quickly:

3 435 1

3 4353 13 Commutative and associative properties 4351 Multiplicative inverse property 435 Multiplicative identity property c) Use the distributive property to rewrite this expression.

6 284 28(64)28 10 28 280

Now do Exercises 1–16

Combining like terms

Use the distributive property to perform the indicated operations.

a) 3x5x b) 5xy(4xy)

Solution

a) 3x5x(35)x Distributive property 8x Add the coefficients.

Because the distributive property is valid for any real numbers, we have 3x5x8x no matter what number is used for x.

b) Since the distributive property is valid also for subtraction, abaca(bc), we can remove xy from the two terms.

5xy(4xy)[5(4)]xy Distributive property

1xy 5(4) 54 1

xy Multiplying by 1 is the same as taking the opposite.

Now do Exercises 17–22

E X A M P L E 2

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Of course, we do not want to write out all of the steps shown in Example 2 every time we combine like terms. We can combine like terms as easily as we can add or subtract their coefficients.

E X A M P L E 3 Combining like terms Perform the indicated operations.

a) w2w b) 3a(7a) c) 9x5x

d) 7xy(12xy) e) 2x24x2 f ) 1

2x 1 4x

Solution

a) w2w1w2w3w b) 3a(7a) 10a

c) 9x5x 4x d) 7xy(12xy)19xy

e) 2x24x26x2 f) 1

2x 1

4x 12 14x 14x

Now do Exercises 23–36

There are no like terms in expressions such as

25x, 3xy5y, 3w5a, and 3z25z.

The terms in these expressions cannot be combined.

U 3 V Products and Quotients

To simplify an expression means to perform operations, combine like terms, and get an equivalent expression that looks simpler. However, simplify is not a precisely defined term. An expression that uses fewer symbols is usually considered simpler, but we should not be too picky with this idea. Simplifying 2x3x we get 5x, but we would not say that

xis simpler than1

2x. Some would say that 2ax2ay is simpler than 2a(xy) because the parentheses have been removed. However, there are seven symbols in each expression, and five operations indicated in 2ax2ay with only three in 2a(xy). If you are asked to write 2a(xy) as a sum or to remove the parentheses rather than to simplify it, then the answer is clearly 2ax2ay.

In Example 4 we use the associative property of multiplication to simplify some products.

CAUTION

E X A M P L E 4 Finding products Simplify.

a) 3(5x) b) 22x

c) (4x)(6x) d) (2a)(4b)

1-69 1.8 Using the Properties to Simplify Expressions 69

E X A M P L E 5 Finding products quickly Find each product.

a) (3)(4x) b) (4a)(7a) c) (3a)b3 d) 6 2x

Solution

a) 12x b) 28a2 c) ab d) 3x

Now do Exercises 47–52

Solution

a) 3(5x)(3 5)x Associative property of multiplication (15)x Multiply.

15x Remove unnecessary parentheses.

b) 22x212 x Multiplying by 1

2is the same as dividing by 2.

2 12x Associative property of multiplication 1 x Multiplicative inverse property x Multiplicative identity property

c) (4x)(6x)4 6 x x Commutative and associative properties 24x2 Definition of exponent

d) (2a)(4b) 2 4 a b 8ab

Now do Exercises 37–46

Note that 2

xis equivalent to 1

2x in Example 4(b) because division is defined as multiplication by the reciprocal of the divisor. In general,

xis equivalent to 1 b x.

Be careful with expressions such as 3(5x) and 3(5x). In 3(5x), we multiply 5 by 3 to get 3(5x)15x. In 3(5x), both 5 and x are multiplied by the 3 to get 3(5x)153x.

In Example 4 we showed how the properties are used to simplify products.

However, in practice we usually do not write out any steps for these problems—we can write just the answer.

CAUTION

In Section 1.2 we found the quotient of two numbers by inverting the divisor and then multiplying. Since aba1

b, any quotient can be written as a product.

E X A M P L E 6 Simplifying quotients Simplify.

a) 1 5

0x b) 4x

2 8

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It is not correct to divide only one term in the numerator by the denominator. For example,

4 2

7 27

because4 2

7121and 279.

U 4 V Removing Parentheses

In Section 1.7 we used the distributive property to multiply a sum or difference by 1 and remove the parentheses. For example,

1(a 5) a 5 and 1(x 2) x 2.

If 1 is replaced with a negative sign, the parentheses are removed in the same manner because multiplying a number by 1 is equivalent to finding its opposite. So,

(a 5) 1(a 5) a 5 and (x 2) 1(x 2) x 2.

If a subtraction sign precedes the parentheses, it is removed in the same manner also, because subtraction is defined as addition of the opposite. So,

3a(a5)3aa52a5 and 5x(x2)5xx26x2.

If parentheses are preceded by a negative sign or a subtraction symbol, the signs of all terms within the parentheses are changed when the parentheses are removed.

CAUTION

UCalculator Close-Up V

A negative sign in front of parentheses changes the sign of every term inside the parentheses.

Solution

a) Since dividing by 5 is equivalent to multiplying by1

5, we have 1

5 0x 1

5(10x)15 10x(2)x2x.

Note that you can simply divide 10 by 5 to get 2.

b) Since dividing by 2 is equivalent to multiplying by1

2, we have 4x

2 8

1 2(4x8) 1

2 4x 1

2 8 Distributive property 2x4.

Note that both 4 and 8 are divided by 2. So we could have written 4x

2 8

4 2

x 8

2 2x4 or 4x 2 8

2(2x 24)

2x 4.

Now do Exercises 53–64

Some parentheses are used for emphasis or clarity and are unnecessary. They can be removed without changing anything. For example,

(2x 3) (x 4) 2x 3 x 4 3x 1.

In Example 8, we simplify more algebraic expressions, some of which contain unnecessary parentheses.

1-71 1.8 Using the Properties to Simplify Expressions 71

E X A M P L E 7

E X A M P L E 8

Removing parentheses with opposites and subtraction Remove the parentheses and combine the like terms.

a) (x 4) 5x 1 b) (5 y) 2y 6

c) 10 (x 3) d) 3x 6 (2x 4)

Solution

The procedure is the same for each part: change the signs of each term in parentheses and then combine like terms.

a) (x 4) 5x 1 x 4 5x 1 4x 3

b) (5 y) 2y 6 5 y 2y 6 3y 1

c) 10 (x 3) 10 x 3 x 7

d) 3x 6 (2x 4) 3x 6 2x 4 x 2

Now do Exercises 65–80

Simplifying algebraic expressions Simplify each expression.

a) (2x 3) (5x 7) b) (3x 6x) 5(4 2x) c) 2x(3x 7) 3(x 6) d) x 0.02(x 500)

Solution

a) (2x 3) (5x 7) 2x 3 5x 7 Remove unnecessary parentheses.

3x 4 Combine like terms.

b) (3x 6x) 5(4 2x) 3x 6x 20 10x Distributive property

7x 20 Combine like terms.

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E X A M P L E 9 Perimeter of a rectangle

Find an algebraic expression for the perimeter of the rectangle shown here and then find the perimeter if x 15 inches.

Solution

The perimeter of any figure is the sum of the lengths of its sides:

2(x) 2(2x 1) 2x 4x 2 6x 2

So 6x 2 is an algebraic expression for the perimeter. If x 15 inches, then the perimeter is 6(15) 2 or 92 inches.

Now do Exercises 115–118

2x 1

x x

U 5 V Applications

Warm-Ups ▼

Fill in the blank.

1. An expression containing a number or the product of a number and one or more variables raised to powers is a

2. terms are terms with the same variables and the same exponents.

3. The number preceding the variables(s) in a term is the .

4. To an expression we combine like terms and perform operations to get an equivalent expression that looks simpler.

5. Multiplying a number by 1 changes the of the number.

True or false?

6. The expressions 3x2y and 5xy2are like terms.

7. The coefficient in 7ab3is –7.

8. The expression 62(x6) simplified is 2x18.

9. 1(x – 4) x4 for any real number x.

10. (3a)(4a)12a for any real number a.

11. bbb2for any real number b.

12. 3(52)15 6

c) 2x(3x 7) 3(x 6) 6x2 14x 3x 18 Distributive property 6x2 11x 18 Combine like terms.

d) x 0.02(x 500) 1x 0.02x 10 Distributive property

0.98x 10 Combine like terms.

Now do Exercises 81–98

U1V Using the Properties in Computation Use the appropriate properties to evaluate the expressions.

See Example 1.

1. 35(200) 2. 15(300)

3. 4

3(0.75) 4. 5(0.2)

5. 2567822 6. 1288376

7. 35 335 7 8. 98 4782 478

9. 18 42 1

4 10. 19 3 2 1

11. (120)(300) 12. 150 200

13. 12 375(66) 14. 3542(2 48) 15. 786842

16. 47126126

U2V Combining Like Terms

Combine like terms where possible. See Examples 2 and 3.

17. 5w6w 18. 4a10a

19. 4xx 20. a6a

21. 2x(3x) 22. 2b (5b)

23. 3a (2a) 24. 10m(6m)

25. aa 26. a a

27. 10 6t 28. 9 4w

29. 3x25x2 30. 3r24r2

31. 4x2x2 32. 6w2w

33. 5mw212mw2 34. 4ab2 19ab2 35. 1

3a 1

2a 36. 3

5b b

U3V Products and Quotients

Simplify the following products or quotients. See Examples 4–6.

37. 3(4h) 38. 2(5h)

39. 6b(3) 40. 3m(1)

41. (3m)(3m) 42. (2x)(2x)

43. (3d)(4d ) 44. (5t)(2t) 45. (y)(y) 46. y(y)

47. 3a(5b) 48. 7w(3r)

49. 3a(2b) 50. 2x(3y)

51. k(1k) 52. t(t1)

53. 3 3

y 54.

9 9t 55. 1

5y 56. 1 2

2b 57. 22

y 58. 6m3

59. 8y4

y 60. 1025

61. 6a 3 3

62. 8x 2

6 63. 9

3 6

64. 10

5 5x

U4V Removing Parentheses

Simplify each expression by removing the parentheses and com- bining like terms. See Example 7.

65. (5x 1) 7x 66. (7a 3) 8a 67. (c 4) 5c 9 68. (y 4) 9 4y 69. (7b 2) 1 70. (a 1) 9 71. (w 4) 8 w 72. (y 3) 9y 1

Exercises

UStudy Tips V

• When you get a test back, don’t simply file it in your notebook. Rework all of the problems that you missed.

• Being a full-time student is a full-time job. A successful student spends 2 to 4 hours studying outside of class for every hour spent in the classroom.

1.8

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73. x(3x1) 74. 4x(2x5) 75. 5( y3) 76. 8(m6) 77. 2m3(m9)

78. 78t(2t6) 79. 3(w2) 80. 5x(2x9)

Simplify the following expressions by combining like terms.

See Example 8.

81. 3x5x69 82. 2x6x715 83. (2x3)(7x4) 84. (3x12)(5x9) 85. 3a7(5a6) 86. 4m5(m2) 87. 2(a4)3(2a) 88. 2(w6)3(w5) 89. 3x(2x3) 5(2x3) 90. 2a(a5) 4(a5) 91. b(2b1) 4(2b1) 92. 2c(c8) 3(c8) 93. 5m6(m3)2m 94. 3a2(a5)7a 95. 53(x2)6 96. 72(k3)k6 97. x0.05(x10) 98. x0.02(x300) Simplify each expression.

99. 3x(4x) 100. 28x11x 101. y5(y9) 102. a(bca) 103. 7(82ym) 104. x8(3x) 105. 1

2(102x) 1 3(3x6) 106. 1

2(x20) 1 5(x15) 107. 1

2(3a1) 1 3(a5) 108. 1

4(6b2) 2

3(3b2)

109. 0.2(x3)0.05(x20) 110. 0.08x0.12(x100) 111. 2k13(5k6)k4 112. 2w33(w4)5(w6) 113. 3m3[2m3(m5)]

114. 6h4[2h3(h9)(h1)]

U5V Applications

Solve each problem. See Example 9.

115. Perimeter of a corral. The perimeter of a rectangular corral that has width x feet and length x40 feet is 2(x)2(x40). Simplify the expression for the perimeter. Find the perimeter if x30 feet.

Figure for Exercise 115 x 40 ft x ft

116. Perimeter of a mirror. The perimeter of a rectangular mirror that has a width of x inches and a length of x16 inches is 2(x)2(x16) inches. Simplify the expression for the perimeter. Find the perimeter if x14 inches.

117. Married filing jointly. The value of the expression 93500.25(x67,900)

is the 2009 federal income tax for a married couple filing jointly with a taxable income of x dollars, where x is over

$67,900 but not over $137,050 (Internal Revenue Service, www.irs.gov).

a) Simplify the expression.

b) Use the expression to find the amount of tax for a couple with a taxable income of $80,000.

c) Use the accompanying graph to estimate the 2009 federal income tax for a couple with a taxable income of $200,000

d) Use the accompanying graph to estimate the taxable income for a couple who paid $80,000 in federal income tax.

1-75 1.8 Using the Properties to Simplify Expressions 75

118. Marriage penalty eliminated. The value of the expression 46750.25(x33,950)

is the 2009 federal income tax for a single taxpayer with taxable income of x dollars, where x is over $33,950 but not over $82,250.

a) Simplify the expression.

b) Find the amount of tax for a single taxpayer with taxable income of $40,000.

c) Who pays more, a married couple with a joint taxable income of $80,000 or two single taxpayers with

taxable incomes of $40,000 each? See Exercise 117.

Getting More Involved 119. Discussion

What is wrong with the way in which each of the following expressions is simplified?

a) 4(2x)8x b) 4(2x)84x32x

c) 2x

d) 5(x3)5x32x

120. Discussion

An instructor asked his class to evaluate the expression 12x for x5. Some students got 0.1;

others got 2.5. Which answer is correct and why?

4x 2 Figure for Exercise 117

Tax (thousands of $)

100 120

0 20 40 60 80

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Wrap-Up

Summary

The Real Numbers Examples

Counting or natural numbers 1, 2, 3, . . . Whole numbers 0, 1, 2, 3, . . .

Integers . . . , 3, 2, 1, 0, 1, 2, 3, . . .

Rational numbers aba and b are integers with b0 3

2, 5,6, 0 Irrational numbers xx is a real number that is not rational 2, 3, Real numbers The set of real numbers consists of all

rational numbers together with all irrational numbers.

Intervals of real numbers If a is less than b, then the set of real The notation (1, 9)

numbers between a and b is written represents the real numbers as (a, b). The set of real numbers between 1 and 9.

between a and b inclusive is written The notation [1, 9]

as [a, b]. represents the real numbers

between 1 and 9 inclusive.

Fractions Examples

Reducing fractions

Building up fractions

Multiplying fractions

Dividing fractions

Adding or subtracting fractions

1 5 2 5 3 5 ac

b c

b a b

3 5 2 5 1 5 ac

b c

b a b

5 6 10 12 5 4 2 3 4 5 2 3 d

c a b c d a b

8 15 4 5 2 3 ac

bd c d a b

15 40 35 85 3

8 ac

bc a

2 3 22 23 4

6 a

b ac bc

Chapter

1-77 Chapter 1 Summary 77

Least common denominator The smallest number that is a

multiple of all denominators.

Operations with Real Numbers Examples

Absolute value a

Sum of two numbers with Add their absolute values. The sum has 3(4) 7 like signs the same sign as the given numbers.

Sum of two numbers with Subtract the absolute values of the 473 unlike signs (and different numbers. The answer is positive if the

absolute values) number with the larger absolute value is positive.

The answer is negative if the number 74 3 with the larger absolute value is

negative.

Sum of opposites The sum of any number and its opposite is 0. 660

Subtraction of signed aba(b) 353(5) 2

numbers Subtract any number by adding its opposite. 4(3)437

Product or quotient Like signs↔ Positive result (3)(2)6

Unlike signs↔ Negative result (8)2 4

Definition of exponents For any counting number n, 232228

anaaa. . .a. (5)225

n factors 52 (52) 25

Order of operations No parentheses or absolute value present:

1. Exponential expressions 52313

2. Multiplication and division 23517 3. Addition and subtraction 453249 With parentheses or absolute value:

First evaluate within each set of (23)(57) 10 parentheses or absolute value, using the 232511 order of operations.

Properties of the Real Numbers Examples

For any real numbers a, b, and c Commutative property of

Addition abba 5775

Multiplication abba 6336

33, 00 33 a if a is positive or zero

a if a is negative

5 12 2 12 3 12 1 6 1 4

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Associative property of

Addition a(bc)(ab)c 1(23)(12)3

Multiplication a(bc)(ab)c 2(34)(23)4

Distributive properties a(bc)abac 2(3x)62x

a(bc)abac 2(x5) 2x10

Additive identity property a0a and 0aa 50055

Zero is the additive identity.

Multiplicative identity property 1 aa and a1a 7 11 77 One is the multiplicative identity.

Additive inverse property For any real number a, there is a number 3(3)0 a (additive inverse or opposite) such that 330 a(a)0 and aa0.

Multiplicative inverse property For any nonzero real number a there is 3 1 a number 1

a(multiplicative inverse or reciprocal) such that

a 1 and a1. 31

Multiplication property of 0 a 00 and 0 a0 5 00

0(7)0 1

3 1

a 1

1 3

9. A is a rational number that is not an integer.

10. A fraction is by dividing out common factors of the numerator and denominator.

11. A fraction is in terms if the numerator and denominator have no common factors.

12. If a is a real number, then a is the inverse of a.

13. The of operations is the order in which operations are to be performed in the absence of grouping symbols.

14. The least common multiple of the denominators is the common denominator.

15. The value of a number is its distance from 0 on the number line.

16. The number 0 is the identity.

17. The number 1 is the identity.

18. In the division a bc, b is the and c is the .

Enriching Your Mathematical Word Power Fill in the blank.

1. The numbers {. . . ,3,2,1, 0, 1, 2, 3, . . .} are the .

2. The numbers {1, 2, 3, 4, . . .} are the or counting numbers.

3. The numbers {0, 1, 2, 3, 4, . . .} are the numbers.

4. The real numbers that can be expressed as a ratio of two integers are the numbers.

5. The real numbers that cannot be expressed as a ratio of two integers are the numbers.

6. A is an expression containing a number or the product of a number and one or more variables raised to powers.

7. Terms that have the same variables with the same exponents are terms.

8. A letter that is used to represent some numbers is a .

1-79 Chapter 1 Review Exercises 79

20. A(n) fraction has a larger numerator than denominator.

19. A(n) number is a natural number 2 or larger that has no factors other than itself and 1.

Review Exercises

1.1 The Real Numbers

Which of the numbers 5,2, 0, 1, 2, 3.14, , and 10 are

1. whole numbers?

2. natural numbers?

3. integers?

4. rational numbers?

5. irrational numbers?

6. real numbers?

True or false? Explain your answer.

7. Every whole number is a rational number.

8. Zero is not a rational number.

9. The counting numbers between 4 and 4 are 3, 2, 1, 0, 1, 2, and 3.

10. There are infinitely many integers.

11. The set of counting numbers smaller than the national debt is infinite.

12. The decimal number 0.25 is a rational number.

13. Every integer greater than 1 is a whole number.

14. Zero is the only number that is neither rational nor irrational.

Graph each set of numbers.

15. The set of integers between 3 and 3

16. The set of natural numbers between 3 and 3

17. The set of real numbers between 1 and 4

18. The set of real numbers between 2 and 3 inclusive

Write the interval notation for each interval of real numbers.

19. The set of real numbers between 4 and 6 inclusive 20. The set of real numbers greater than 2 and less than 5 21. The set of real numbers greater than or equal to 30 22. The set of real numbers less than 50

1.2 Fractions

Perform the indicated operations.

23. 1 33

8 24. 2

4 25. 3

5 10 26. 3

5 10 27. 2

5 1 1 5

4 28. 7 1

2 29. 42

3 30.

1 7

21 4

31. 32. 9

1.3 Addition and Subtraction of Real Numbers Evaluate.

33. 57 34. 9(4)

35. 3548 36. 3 9

37. 125 38. 12 5

39. 12(5) 40. 9 (9)

41. 0.0512 42. 0.03 (2)

43. 0.1(0.05) 44. 0.3 0.3

45. 46.

47. 48.

1.4 Multiplication and Division of Real Numbers Evaluate.

49. (3)(5) 50. (9)(4)

51. (8)(2) 52. 50 (5) 1 4 1 3 2

5 1 3

1 4 2 3 1

2 1 3

3 4 1

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53. 54.

55. 1213 56. 8 13

57. 0.090.3 58. 4.2 (0.3)

59. (0.3)(0.8) 60. 0 (0.0538)

61. (5)(0.2) 62. (12)