CHAPTER 5 EXPONENTS AND POLYNOMIALS
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1 Chapter Five Additional Eercises 1 CHAPTER EXPONENTS AND POLYNOMIALS Section.1 The Product Rule and Power Rules for Eponents Objective 1 Use eponents. Write the epression in eponential form and evaluate ( 1( 1( 1( 1( 1( Write the epression in eponential form. 8. ( ( ( 7. mmmmmm 9. ( ef ( ef ( ef ( ef 9. (.st(.st(.st Evaluate the eponential epressions. Name the base and the eponent. 11. ( ( ( Objective Use the product rule for eponents. Use the product rule to simplif the epression, if possible. Write the answer in eponential form ( 7 ( ( (
2 16 Chapter Five Additional Eercises Multipl.. p 6p 6. k 16k 7. 6a 1 a 8. 1b( 1b ( m ( 7m. ( ( In the following eercises, add the given terms. Then start over and multipl them. 1..,. a, 6 a, a., t,6 t, 7t Objective Use the rule ( a m n mn = a. Simplif the epression. Write the answer in eponential form.. ( 6. ( ( ( 1 9. ( 8. ( ( 11. ( 7. ( 1 1. ( ( ( 1 7
3 Chapter Five Additional Eercises 17 ab = a b. Objective Use the rule ( m m m Simplif the epression. 7. ( a 8. ( z 9. ( pq. ( rs 1. ( ab 7. ( w z r t. ( qr 7. (. ( r s 6. cd ( 7. ( rs 6 8. ( c d Objective Use the rule a b m a = b m m. Simplif the epression. Assume all variables represent nonzero real numbers a b 6. w 6. z z 66. z a b 69. r s 7. 1 k Objective 6 Use combinations of rules. 71. ( a ( a 7. ( a bc ( abc 6 z z 7. ( m 7. ( ( 7 7. ( ( ab 77. ( z ( z 78. ( (
4 18 Chapter Five Additional Eercises Mied Eercises Evaluate the eponential epression. Name the base and the eponent. 79. ( ( ( Simplif the epression. Write the answer in eponential form. 8. ( ( a b 88. ( 87. ( ( 7 9. ( 9st ( 9st 6 9w 91. ( rst 9. ( n n 7 9. ( m ( m 9. ( 6q ( 6q b ab km p n 98. ( n. ( 7 z ( 99. ( pq ( pq
5 Chapter Five Additional Eercises 19 Writing/Conceptual Eercises 1. Eplain how ( 6 and. Eplain how ( and 6 are different. are different.. Eplain wh the product rule for eponents does not appl to the epression. Then evaluate the epression b finding the individual powers and multipling the results.. An algebra student applied the product rule for eponents in the following wa: ( ( = 1 1. Eplain to the student wh this is not correct and show how to obtain the correct answer.. After listening to our eplanation, the student in Eercise # said, Now I understand. Let me tr another one. This time the student applied the product rule in the following wa: ( ( = Eplain to the student wh this is also incorrect, and then show how to obtain the correct answer. 6. Eplain wh ( 7. Is ( equivalent to ( a b is not equivalent to ( ab 6 8 or? Eplain our answer ab. 8. On a math test covering the material in this chapter, a student was asked to simplif the following epressions. ab c 1. ( c. ( a bc She wondered wh the first epression was written with restriction c, while the second was not, even though the epressions contain the same numbers and variables. Eplain the reason for this.
6 Chapter Five Additional Eercises Section. Integer Eponents and the Quotient Rule Objective 1 Use as an eponent. Evaluate the epression ( ( ( + ( ( ( 119. w ( r. Objective Use negative numbers as eponents. Evaluate the epression ( 1. ( Simplif b using the definition of negative eponents. Write the epression with onl positive eponents. Assume all variables represent nonzero real numbers.. r r 1. r
7 Chapter Five Additional Eercises 11 Objective Use the quotient rule for eponents. Use the quotient rule to simplif the epression. Write answers with onl positive eponents. Assume that all variables represent nonzero real numbers ( ( 19. ( ( km 7 km m m p b c p b c q 19. e f. e f Objective Use combinations of rules. Use a combination of the rules for eponents to simplif the epression. Write answers with onl positive eponents. Assume that all variables represent nonzero real numbers. 11. ( ( a a a 1. ( 16. ( 17. ( w ( w ( r s ( r s 19. ( m ( m 16. ( 9 ( 9 6
8 1 Chapter Five Additional Eercises 161. ( t ( t 6 ( t 16. ( q ( q ( q k t kt Mied Eercises Evaluate the epression. 16. ( ( 168. ( 169. ( ( Simplif the epression. Write answers with onl positive eponents. Assume that all variables represent nonzero real numbers a a ( ( ( k k ( ( ( ( z 8 z z z
9 Chapter Five Additional Eercises ( 1 1 z ( ( z ( m n p 1 ( pm Writing/Conceptual Eercises Decide whether the epression is positive, negative, or zero ( ( ( If one side of a square measures 197. If one edge of a cube measures centimeters, what is the area of the square? feet, what is the volume of the cube? On an algebra quiz, a student evaluated the epression + as follows: = = =. 16 Eplain wh this is incorrect and show the correct solution. Section. An Application of Eponents: Scientific Notation Objective 1 Epress numbers in scientific notation. Write the number in scientific notation ,. 69, ,, ,7..1
10 1 Chapter Five Additional Eercises Objective Convert numbers in scientific notation to numbers without eponents. Write the numbers without eponents Objective Use scientific notation in calculations. Perform the indicated operations with the numbers in scientific notation, and then write the answer without eponents.. ( ( 7. (. ( (. (.1 6. ( ( ( ( ( (. ( 6 ( ( (.8 ( ( 7 ( ( 7. (. ( 6 (.
11 Chapter Five Additional Eercises 1 Mied Eercises If a number is written without eponents, rewrite it in scientific notation. If a number is written in scientific notation, rewrite it without eponents..,9, Perform the indicated operations with the numbers in scientific notation, and write the answer without eponents (. ( (. ( ( ( 6. ( 6 ( 6 ( ( 6 ( ( 8 ( ( ( 8. 1 ( 6 ( 9. 6 ( 7. (.8 ( 1.8 ( ( 1. (.9 8 ( 7 (.
12 16 Chapter Five Additional Eercises Writing/Conceptual Eercises Determine whether the given number is written in scientific notation, as defined in the tetbook. If it is not, write it as such Eplain in our own words some reasons wh ou think scientists prefer to work with numbers that are in scientific notation rather than numbers written without eponents. 7. Wh do ou think that was chosen as the base for all numbers written in scientific notation? Section. Adding and Subtracting Polnomials; Graphing Simple Polnomials Objective 1 Identif terms and coefficients. For the polnomial, determine the number of terms and name the coefficients of the terms b a a 7. f + f f w z + z a.b
13 Chapter Five Additional Eercises 17 Objective Add like terms. In the polnomial, add like terms whenever possible. Write the result in descending powers of the variable s + 8s 8. t + ( 7t m.9m 86. 8c + 11c c c c + c 88. 8a a + a 7a r r+ r + r r.8r 6.r 9.7r m + m m + m Objective Know the vocabular for polnomials. Choose one or more of the following descriptions for each epression: (a polnomial, (b polnomial written in descending order, (c not a polnomial w + 7w + w k a + a 6a + 6 f f f + Simplif the polnomial, if possible, and write the resulting polnomial in descending powers of the variable. Then give the degree of this polnomial, and tell whether it is a monomial, a binomial, a trinomial, or none of these z + z z n n + n m m m p + p p
14 18 Chapter Five Additional Eercises Objective Evaluate polnomials. Find the value of the polnomial when (a = and (b = Objective Add and subtract polnomials. Add m + m m m 1m m +. w w w w w + 8w p + p p 11 p p 8p p z z z z z 8 9 z z + z + z + z. ( ( 7 6. ( ( + 7. ( ( + 6+
15 Chapter Five Additional Eercises ( z 7 6z z ( z 6 + z + z + 9. ( + + ( + +. ( m m + m + m 6 + ( m m m+ Subtract m 6m + m m k k 18 11k k n + n n n z z + z + 8 z z z 9. ( ( 6+. ( ( ( m m+ 7 ( m+ 8. ( 1z z + z ( z z 1. ( 8a a a+ 11 ( a + a a + a. ( p 9p 6p + 8p 1 ( p p 1
16 Chapter Five Additional Eercises Objective 6 Graph equations defined b polnomials of degree. Select several values for ; then find the corresponding -values, and graph.. = 6. = + 7. = 8. = 9. =. = Mied Eercises Perform the indicated operations. Write each resulting polnomial in descending powers of the variable. Then give the degree of this polnomial, and tell whether it is a monomial, a binomial, a trinomial, or none of these. 1. ( (. ( 6 + ( + 1. ( ( ( + + ( + 8. ( p + p 6p + 7p 1 ( p p ( ( a a+ + ( a + a 1 + ( a + 8. ( + ( + ( For the following polnomials, state the coefficient of the second term. Then evaluate the polnomial when = and =
17 Chapter Five Additional Eercises 11 Add or subtract as indicated. 67. ( ( ( c cd + 6d ( d + cd c 69. ( rs+ r s ( 6s r sr 7. ( 6ab + bc ac + ( ca cb 8ba 71. ( ( ( + 7. (.6ab +.a.b (.8a +.b +.6ab Add r t + rt 8rt r t rt 6rt Subtract ab 6ab + ab + 6ab 7ab 76. 6rs + 7rt st rs 9rt st Writing/Conceptual Eercises 77. Eplain wh the degree of the term is not. What is its degree? 78. Can the sum of two polnomials in, both of degree, be of degree? If so, give an eample. 79. Can the sum of two polnomials in, both of degree, be of degree 1? If so, give an eample. 8. Is it possible to add two trinomials and obtain a sum which is a binomial? If so, give an eample.
18 1 Chapter Five Additional Eercises Section. Multipl Polnomials Objective 1 Multipl a monomial and a polnomial. Find the product. 81. ( ( 8. ( 9 ( 8 8. z( z p ( 8p 7p 7 8. k( k + k k + 6k ( k( k+ k 88. 7m( m + m 89. a( a+ 7a 9. r ( r 7r mn( m mn + 7n 9. rs ( rs 6rs+ rs Objective Multipl two polnomials. Find the product. 9. ( + ( + 9. ( 6( 9. ( p+ ( p+ 96. ( 9+ a( a 97. ( n+ 1( n ( ( ( 9r 7( r +. ( ( ( + ( +. ( + 1( +. ( 1( + 7. ( m + 1( m m m. ( z ( z z + z z ( + + 1( +
19 Chapter Five Additional Eercises 1 Objective Multipl binomials b the FOIL method. Use the FOIL method to find the product. 7. ( ( + 8. ( + 6( + 9. ( r 8( r+ 7. ( z ( z ( k 1( k+ 1. ( m ( m ( 8+ ( ( + ( 1. ( m+ n( m n 16. ( p+ q( p q 17. ( m( + m 18. ( k 11( k ( 1+ (. ( 8a( 1 6a 1. ( ( 1+. ( 7p+ 1( p 1. ( r s( r+ s. ( 1m+ n( m+ n. ( + ( 6. ( v + w ( v w Mied Eercises Find the product. 7. ( ( 8. m( m 7 9. ( q 7( q 6. ( 9p+ r( 6p r 1. ( 6( p ( 1+ 6 p 7 p + p. ( 6 8 ( ( 8 6(. ( ( ( + + 1( +
20 1 Chapter Five Additional Eercises 7. ( + 1( ( 1( ( a 1( a + a a. ( b 1( b b + b b 1 1. ( + ( +. ( + ( ( m + m ( m m+ 1. ( ( + Section.6 Special Products. Objective 1 Square binomials. Find the square b using the pattern for the square of a binomial.. ( z + 6. ( t 9 7. ( ( 1 9. ( m 7. ( p (. ( (. ( w q m 9p. 1 z a b Objective Find the product of the sum and difference of two terms. Find the product b using the pattern for the sum and difference of two terms. 9. ( z 8( z ( k 1( k ( b 7( b ( 8k p( 8k+ p
21 Chapter Five Additional Eercises 1 6. ( p+ q( p q 6. ( + k( k 6. ( 9 ( m m a+ b 6a b 7. s+ t s t 71. ( + ( 7. ( m n ( m + n Objective Find greater powers of binomials. Find the product. 7. ( 1 7. ( r ( ( ( ( k ( t 8. ( ( z 6 8. ( s+ t 8. ( 8. ( b 1 Mied Eercises Find the product. 8. ( 86. ( 6b 11( 6b ( b 88. ( 7t+ t( 7t u 89. ( 9k+ m 9. ( 16w( + 16w
22 16 Chapter Five Additional Eercises 9. 1 j+ k 9. t+ 9u t 9u b c ( 98. ( a b ( 6 ( 6+. (. 1 a b. ( Writing/Conceptual Eercises 7. Eplain how the epression and ( differ. 8. A student is asked to find the product (, and gives the answer numerical eample to eplain wh this is incorrect. 16. Use a 9. A student remembers that the square of a binomial is a trinomial. When asked to find, he gives the answer Eplain wh this answer is incorrect. (. Based on our eperience in finding powers of binomials such as ( a+ b, ( a+ b, and ( a b, ( a b + how man terms would ou epect to find in the simplified answer for +? Eplain our answer. (Do not actuall find the product.
23 Chapter Five Additional Eercises 17 Section.7 Dividing Polnomials Objective 1 Divide a polnomial b a monomial. Divide the polnomial b m m + m 1. 18m 9m m + m m 16. 7m + m 6m 18. 6m m +. 1m 9m + 6m m m m m m m 1m m m+ Perform the division. 1. 8p + p p ( (. ( 9z 7z + z 11 ( z. ( 6 ( 6. ( 6m m + m ( m 7. ( m 7m 1 ( 7m + 8. p p p p q q q q z z z z z
24 18 Chapter Five Additional Eercises Objective Divide a polnomial b a polnomial. Perform the division p + 7p p 9. ( + + ( +. ( r r ( r 1. ( b 1b+ ( b+ 1. ( 9w 6w+ 1 ( w 1. w w+ w. 6b + 7b+ 7 b + 7. m m+ 9 m z z z z m m m m ( p p 9 p + 18p 8 ( p. ( ( b 1 b 1
25 Chapter Five Additional Eercises Writing/Conceptual Eercises 61. Suppose that a polnomial in the variable has degree and it is divided b a monomial in the variable having degree. Describe the quotient in mathematical terms, giving tpe of epression and the degree. 6. In her algebra class, Mischa volunteered to work the following division problem on the blackboard: 1z 9z + z. z Mischa s answer was z z. She said that the z terms in the numerator and denominator would cancel out. Was she correct? Eplain. 6. Matt, who is one of Mischa s classmates, said that her answer could not possibl be correct because it onl had terms. Was he right? 6. Stephanie, who is another one Mischa s classmates, told Mischa she could have found her mistake if she had checked her answer. Show the result if Stephanie checked Mischa s answer and if she checked the correct answer. 6. Justin s algebra instructor put the following division problem on a quiz: ( ( 1 1. Justin wrote the problem in the following wa: 1 1. What difficult do ou epect Justin will have when he tries to perform this division? 66. You are given the following division problem: ( + + ( 1 8. B looking at this problem, but without performing the division, determine which of the following would be the first term of the quotient. (a (b (c (d 9
26 16 Chapter Five Additional Eercises 67. A student performs the division +, and obtains a remainder of. Without working the problem, eplain wh this remainder cannot be correct. 68. Two students give the following answers to a problem in which a polnomial is divided b the binomial : Amanda: Dale: Can both answers be correct? Eplain.
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