Chapter 5.pdf - Chapter 5 Chapter 5 Exponents and Polynomials 5.1 Chapter 5 Part X 1 5.1.1 Addition and Subtraction of Polynomials 5.1.1.1 Overview Like

Chapter 5.pdf - Chapter 5 Chapter 5 Exponents and...

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Unformatted text preview: Chapter 5 Chapter 5: Exponents and Polynomials 5.1 Chapter 5 Part X 1 5.1.1 Addition and Subtraction of Polynomials 5.1.1.1 Overview • • • Like Terms Combining Like Terms Simplifying Expressions Containing Parentheses 5.1.1.2 Like Terms Like Terms Terms whose variable parts, including the exponents, are identical are called like terms . Like terms is an appropriate name since terms with identical variable parts and dierent numerical coecients represent dierent amounts of the same quantity. As long as we are dealing with quantities of the same type we can Simplifying an Algebraic Expression simplied 5.1.1.3 Sample Set A combine them using addition and subtraction. An algebraic expression can be by combining like terms. Combine the like terms. Example 5.1 6 +4 Example 5.2 houses houses = 10 houses. 6 and 4 of the same type give 10 of that type. 6 houses + 4 houses + 2 motels = 10 houses + 2 motels . 6 and 4 of the same type give 10 of that type. Thus, we have 10 of one type and 2 of another type. Example 5.3 Suppose we let the letter x represent "house." Then, 6x + 4x = 10x . 6 and 4 of the same type give 10 of that type. 1 This content is available online at < ;. Available for free at Connexions < ; 257 258 CHAPTER 5. Example 5.4 Suppose we let x represent "house" and y CHAPTER 5: EXPONENTS AND POLYNOMIALS represent "motel." 6x + 4x + 2y = 10x + 2y (5.1) 5.1.1.4 Practice Set A Exercise 5.1.1.1 Like terms with the same numerical coecient represent equal amounts of the same quantity. (Solution on p. 312.) Like terms with dierent numerical coecients represent . 5.1.1.5 Combining Like Terms Since like terms represent amounts of the same quantity, they may be combined, that is, like terms may be added together. 5.1.1.6 Sample Set B Simplify each of the following polynomials by combining like terms. Example 5.5 2x + 5x + 3x. 2x's, then 5 more, 2x + 5x + 3x = 10x There are then 3 more. This makes a total of 10x's. Example 5.6 7x + 8y − 3x. 7x's, we From lose 3x's. This makes 4x's. The 8y 's represent a quantity dierent from the x's and therefore will not combine with them. 7x + 8y − 3x = 4x + 8y Example 5.7 4a3 − 2a2 + 8a3 + a2 − 2a3 . 4a3 , 8a3 , and −2a3 represent quantities of the same type. 4a3 + 8a3 − 2a3 = 10a3 −2a2 and a2 represent quantities of the same type. −2a2 + a2 = −a2 Thus, 4a3 − 2a2 + 8a3 + a2 − 2a3 = 10a3 − a2 5.1.1.7 Practice Set B Exercise 5.1.1.2 4y + 7y Exercise 5.1.1.3 Simplify each of the following expressions. (Solution on p. 312.) (Solution on p. 312.) 3x + 6x + 11x Available for free at Connexions < ; 259 Exercise 5.1.1.4 5a + 2b + 4a − b − 7b Exercise 5.1.1.5 10x − 4x + 3x − 12x + 5x + 2x + x + 8x Exercise 5.1.1.6 3 3 2 3 2 (Solution on p. 312.) (Solution on p. 312.) 3 (Solution on p. 312.) 2a5 − a5 + 1 − 4ab − 9 + 9ab − 2 − 3 − a5 5.1.1.8 Simplifying Expressions Containing Parentheses Simplifying Expressions Containing Parentheses When parentheses occur in expressions, they must be removed before the expression can be simplied. Distributive Property Parentheses can be removed using the distributive property. 5.1.1.9 Sample Set C Example 5.8 Simplify each of the following expressions by using the distributive property and combining like terms. Example 5.9  4x + 9 x2 − 6x − 2 + 5 2 4x + 9x − 54x − 18 + 5 Remove parentheses. Combine like terms. 2 −50x + 9x − 13 By convention, the terms in an expression are placed in descending order with the highest degree term appearing rst. Numerical terms are placed at the right end of the expression. The commutative property of addition allows us to change the order of the terms. 9x2 − 50x − 13 Example 5.10 2 + 2 [5 + 4 (1 + a)] Eliminate the innermost set of parentheses rst. 2 + 2 [5 + 4 + 4a] By the order of operations, simplify inside the parentheses before multiplying (by the 2). 2 + 2 [9 + 4a] Remove this set of parentheses. 2 + 18 + 8a Combine like terms. 20 + 8a Write in descending order. 8a + 20 Example 5.11 x (x − 3) + 6x (2x + 3) Use the rule for multiplying powers with the same base. Available for free at Connexions < ; 260 CHAPTER 5. x2 − 3x + 12x2 + 18x CHAPTER 5: EXPONENTS AND POLYNOMIALS Combine like terms. 2 13x + 15x 5.1.1.10 Practice Set C Exercise 5.1.1.7  4 (x + 6) + 3 2 + x + 3x − 2x Exercise 5.1.1.8  7 x + x − 4x − x + 1 + 4 x − 2x + 7 Exercise 5.1.1.9 5 (a + 2) + 6a − 7 + (8 + 4) (a + 3a + 2) Exercise 5.1.1.10 x (x + 3) + 4x + 2x Exercise 5.1.1.11   a a + a + 5 + a a + 3a + 4 + 1 Exercise 5.1.1.12 2 [8 − 3 (x − 3)] Exercise 5.1.1.13   Simplify each of the following expressions by using the distributive property and combining like terms. 2 3 (Solution on p. 312.) 2 3 2 (Solution on p. 312.) 3 (Solution on p. 312.) (Solution on p. 312.) 2 3 2 4 (Solution on p. 312.) 2 (Solution on p. 312.) (Solution on p. 312.) x2 + 3x + 7 x + 4x2 + 3 x + x2 Available for free at Connexions < ; 261 5.1.1.11 Exercises Exercise 5.1.1.14 x + 3x Exercise 5.1.1.15 4x + 7x Exercise 5.1.1.16 9a + 12a Exercise 5.1.1.17 5m − 3m Exercise 5.1.1.18 10x − 7x Exercise 5.1.1.19 7y − 9y Exercise 5.1.1.20 6k − 11k Exercise 5.1.1.21 3a + 5a + 2a Exercise 5.1.1.22 9y + 10y + 2y Exercise 5.1.1.23 5m − 7m − 2m Exercise 5.1.1.24 h − 3h − 5h Exercise 5.1.1.25 a + 8a + 3a Exercise 5.1.1.26 7ab + 4ab Exercise 5.1.1.27 8ax + 2ax + 6ax Exercise 5.1.1.28 3a + 6a + 2a Exercise 5.1.1.29 14a b + 4a b + 19a b Exercise 5.1.1.30 10y − 15y Exercise 5.1.1.31 7ab − 9ab + 4ab Exercise 5.1.1.32 210ab + 412ab + 100a b Exercise 5.1.1.33 5x y + 3x y + 2x y + 1, Exercise 5.1.1.34 For the following problems, simplify each of the algebraic expressions. 2 2 2 (Solution on p. 312.) (Solution on p. 312.) (Solution on p. 312.) (Solution on p. 312.) (Solution on p. 312.) (Solution on p. 312.) (Solution on p. 312.) 2 2 2 4 2 0 (Solution on p. 312.) 4 2 4 2 (Solution on p. 312.) (Solution on p. 312.) (Look closely at the exponents.) y 6= 0 (Look closely at the exponents.) (Solution on p. 312.) 8w2 − 12w2 − 3w2 Available for free at Connexions < ; 262 CHAPTER 5. CHAPTER 5: EXPONENTS AND POLYNOMIALS Exercise 5.1.1.35 6xy − 3xy + 7xy − 18xy Exercise 5.1.1.36 7x − 2x − 10x + 1 − 5x − 3x − 12 + x Exercise 5.1.1.37 21y − 15x + 40xy − 6 − 11y + 7 − 12x − xy Exercise 5.1.1.38 1x + 1y − 1x − 1y + x − y Exercise 5.1.1.39 5x − 3x − 7 + 2x − x Exercise 5.1.1.40 −2z + 15z + 4z + z − 6z + z Exercise 5.1.1.41 18x y − 14x y − 20x y Exercise 5.1.1.42 −9w − 9w − 9w + 10w Exercise 5.1.1.43 2x + 4x − 8x + 12x − 1 − 7x − 1x − 6x + 2 Exercise 5.1.1.44 17d r + 3d r − 5d r + 6d r + d r − 30d r + 3 − 7 + 2 Exercise 5.1.1.45 a + 2a − 4a , a 6= 0 Exercise 5.1.1.46 4x + 3x − 5x + 7x − x , x 6= 0 Exercise 5.1.1.47 2a b c + 3a b c + 4a b − a b c, c 6= 0 Exercise 5.1.1.48 3z − 6z + 8z Exercise 5.1.1.49 3z − z + 3z Exercise 5.1.1.50 6x + 12x + 5 Exercise 5.1.1.51 3 (x + 5) + 2x Exercise 5.1.1.52 7 (a + 2) + 4 Exercise 5.1.1.53 y + 5 (y + 6) Exercise 5.1.1.54 2b + 6 (3 − 5b) Exercise 5.1.1.55 5a − 7c + 3 (a − c) Exercise 5.1.1.56 3 2 2 2 (Solution on p. 312.) 2 3 3 2 2 2 5 4 5 3 0 0 3 2 3 3 (Solution on p. 313.) 4 2 3 0 (Solution on p. 313.) 2 2 4 3 2 (Solution on p. 312.) 3 2 3 4 (Solution on p. 313.) 2 0 0 0 2 2 0 0 2 2 (Solution on p. 313.) 0 3 2 (Solution on p. 313.) 3 (Solution on p. 313.) 3 (Solution on p. 313.) (Solution on p. 313.) (Solution on p. 313.) 8x − 3x + 4 (2x + 5) + 3 (6x − 4) Available for free at Connexions < ; 263 Exercise 5.1.1.57 2z + 4ab + 5z − ab + 12 (1 − ab − z) Exercise 5.1.1.58 (a + 5) 4 + 6a − 20 Exercise 5.1.1.59 (4a + 5b − 2) 3 + 3 (4a + 5b − 2) Exercise 5.1.1.60   10x + 3y 4 + 4 10x + 3y Exercise 5.1.1.61 2 (x − 6) + 5 Exercise 5.1.1.62 1 (3x + 15) + 2x − 12 Exercise 5.1.1.63  1 2 + 9a + 4a + a − 11a Exercise 5.1.1.64   1 2x − 6b + 6a b + 8b + 1 5x + 2b − 3a b Exercise 5.1.1.65 2 (Solution on p. 313.) (Solution on p. 313.) 2 2 (Solution on p. 313.) 2 2 2 (Solution on p. 313.) 2 After observing the following problems, can you make a conjecture about 1 (a + b)? 1 (a + b) = Exercise 5.1.1.66 (Solution on p. 313.) Using the result of problem 52, is it correct to write (a + b) = a + b? Exercise 5.1.1.67   3 2a + 2a + 8 3a + 3a Exercise 5.1.1.68  x (x + 2) + 2 x + 3x − 4 Exercise 5.1.1.69  A (A + 7) + 4 A + 3a + 1 Exercise 5.1.1.70  b 2b + 5b + b + 6 − 6b − 4b + 2 Exercise 5.1.1.71 4a − a (a + 5) Exercise 5.1.1.72  x − 3x x − 7x − 1 Exercise 5.1.1.73 ab (a − 5) − 4a b + 2ab − 2 Exercise 5.1.1.74 xy (3xy + 2x − 5y) − 2x y − 5x y + 4xy Exercise 5.1.1.75 3h [2h + 5 (h + 2)] Exercise 5.1.1.76 2k [5k + 3 (1 + 7k)] Exercise 5.1.1.77 8a [2a − 4ab + 9 (a − 5 − ab)] Exercise 5.1.1.78 2 2 (Solution on p. 313.) 2 2 3 2 (Solution on p. 313.) 2 (Solution on p. 313.) 2 2 2 2 2 (Solution on p. 313.) 2 (Solution on p. 313.) (Solution on p. 313.) 6{m + 5n [n + 3 (n − 1)] + 2n2 } − 4n2 − 9m Available for free at Connexions < ; 264 CHAPTER 5. CHAPTER 5: EXPONENTS AND POLYNOMIALS Exercise 5.1.1.79 5 [4 (r − 2s) − 3r − 5s] + 12s Exercise 5.1.1.80   8{9 b − 2a + 6c (c + 4) − 4c + 4a + b} − 3b Exercise 5.1.1.81 5 [4 (6x − 3) + x] − 2x − 25x + 4 Exercise 5.1.1.82 3xy (4xy + 5y) + 2xy + 6x y + 4y − 12xy Exercise 5.1.1.83   9a b a b − 2a b + 6 − 2a a b − 5a b + 3a b − a b Exercise 5.1.1.84 −8 (3a + 2) Exercise 5.1.1.85 −4 (2x − 3y) Exercise 5.1.1.86   (Solution on p. 313.) 2 2 3 3 7 3 5 2 3 2 2 3 2 7 (Solution on p. 313.) 3 5 12 4 9 3 7 (Solution on p. 313.) (Solution on p. 313.) −4xy 2 7xy − 6 5 − xy 2 + 3 (−xy + 1) + 1 5.1.1.12 Exercises for Review Exercise 5.1.1.87   ( here ) Exercise 5.1.1.88 ( here ) Exercise 5.1.1.89 ( here ) Exercise 5.1.1.90 ( here ) (2a + 5) 3x (2a + 5) Exercise 5.1.1.91   ( here ) 3 5n + 6m − 2 3n + 4m x10 y 8 z 2 x2 y 6 3 2 Simplify 3 −3(4−9)−6(−3)−1 Find the value of . 23 . (Solution on p. 313.) 42x2 y 5 z 3 21x4 y 7 so that no denominator appears. 4 Write the expression 5 How many 's are there in 6 Simplify 2 2 (Solution on p. 314.) ? . 7 5.1.2 Basic Properties of Exponents 5.1.2.1 Overview • • • Exponential Notation Reading Exponential Notation The Order of Operations 2 "Basic 3 "Basic Properties of Real Numbers: Rules of Exponents" < ; Operations with Real Numbers: Multiplication and Division of Signed Numbers" < ; 4 "Basic Operations with Real Numbers: Negative Exponents" < ; 5 "Algebraic Expressions and Equations: Algebraic Expressions" < ; 6 "Algebraic Expressions and Equations: Classication of Expressions and Equations" < ; 7 This content is available online at < ;. Available for free at Connexions < ; 265 5.1.2.2 Exponential Notation 8 In Section here repeated exponential notation we were reminded that multiplication is a description for repeated addition. question is Is there a description for Factors describes repeated multiplication is multiplication? The answer is yes. . In multiplication, the numbers being multiplied together are called A natural The notation that factors . In repeated multiplication, all the factors are the same. In nonrepeated multiplication, none of the factors are the same. For example, Example 5.12 18 · 18 · 18 · 18 Repeated multiplication of 18. All four factors, x·x·x·x·x Repeated multiplication of x. All ve factors, 3·7·a Nonrepeated multiplication. None of the factors are the same. superscript on the factor that is repeated Exponential Notation 18, are the same. x, are the same. exponent Exponential notation is used to show repeated multiplication of the same factor. The notation consists of using a If x is any real number and n . The superscript is called an . is a natural number, then xn = |x · x · {z x · ... · x} n factors of x An exponent records the number of identical factors in a multiplication. Note that the denition for exponential notation only has meaning for natural number exponents. We will extend this notation to include other numbers as exponents later. 5.1.2.3 Sample Set A Example 5.13 7 · 7 · 7 · 7 · 7 · 7 = 76 . The repeated factor is 7. The exponent 6 records the fact that 7 appears 6 times in the multiplication. Example 5.14 x · x · x · x = x4 . The repeated factor is x. The exponent 4 records the fact that x appears 4 times in the multiplication. Example 5.15 (2y) (2y) (2y) = (2y) 3 . The repeated factor is 2y . The exponent 3 records the fact that the factor 2y appears 3 times in the multiplication. Example 5.16 2yyy = 2y 3 . The repeated factor is y. The exponent 3 records the fact that the factor y appears 3 times in the multiplication. Example 5.17 2 3 (a + b) (a + b) (a − b) (a − b) (a − b) = (a + b) (a − b) . The repeated factors are (a + b) and (a − b), (a + b) appearing 2 times and (a − b) appearing 3 times. 8 "Basic Properties of Real Numbers: Properties of the Real Numbers" < ; Available for free at Connexions < ; 266 CHAPTER 5. CHAPTER 5: EXPONENTS AND POLYNOMIALS 5.1.2.4 Practice Set A Exercise 5.1.2.1 a·a·a·a Exercise 5.1.2.2 (3b) (3b) (5c) (5c) (5c) (5c) Exercise 5.1.2.3 2 · 2 · 7 · 7 · 7 · (a − 4) (a − 4) Exercise 5.1.2.4 8xxxyzzzzz CAUTION Write each of the following using exponents. (Solution on p. 314.) (Solution on p. 314.) (Solution on p. 314.) (Solution on p. 314.) It is extremely important to realize and remember that an exponent applies only to the factor to which it is directly connected. 5.1.2.5 Sample Set B Example 5.18 8x3 8 · xxx means the factor x and Example 5.19 3 (8x) not 8x8x8x . The exponent 3 applies only to the factor x since it is only to that the 3 is connected. means to the factor (8x) (8x) (8x) since the parentheses indicate that the exponent 3 is directly connected 8x. Remember that the grouping symbols ( ) indicate that the quantities inside are to be considered as one single number. Example 5.20 34(a + 1) 2 means 34 · (a + 1) (a + 1) since the exponent 2 applies only to the factor 5.1.2.6 Practice Set B Exercise 5.1.2.5 4a Exercise 5.1.2.6 (a + 1). Write each of the following without exponents. (Solution on p. 314.) 3 (Solution on p. 314.) 3 (4a) 5.1.2.7 Sample Set C Example 5.21 Select a number to show that Suppose we choose x 2 (2x) 2x2 . (2x) and 2x2 . is not always equal to to be 5. Consider both 2 2 2x2 (2x) (2 · 5) (10) 100 Notice that 2 (2x) = 2x2 only when 2 2 · 52 2 2 · 25 6= 50 x = 0. Available for free at Connexions < ; (5.2) 267 5.1.2.8 Practice Set C Exercise 5.1.2.7 Select a number to show that 2 (5x) (Solution on p. 314.) is not always equal to 5.1.2.9 Reading Exponential Notation x Base x base Exponent n exponent Power x power x to the nth Power x x n x Squared and x Cubed In n 5x2 . , is the is the The number represented by The term n The symbol is read as " x2 n is called a to the . th power," or more simply as "x to the is often read as "x squared," and x3 nth." is often read as "x cubed." A natural question is "Why are geometric terms appearing in the exponent expression?" The answer for x3 is this: x3 means x · x · x. In geometry, the volume of a rectangular box is found by multiplying the length by the width by the depth. A cube has the same length on each side. If we represent this length by the letter cube is x · x · x, which, of course, is described by x3 . (Can you think of why x2 x then the volume of the is read as x squared?) Cube with =x =x depth = x 3 Volume = xxx = x length width 5.1.2.10 The Order of Operations 9 In Section here we were introduced to the order of operations. It was noted that we would insert another The Order of Operations operation before multiplication and division. We can do that now. operations as you come to 1. Perform all operations inside grouping symbols beginning with the innermost set. 2. Perform all exponential them, moving left-to-right. 3. Perform all multiplications and divisions as you come to them, moving left-to-right. 9 "Basic Properties of Real Numbers: Symbols and Notations" < ; Available for free at Connexions < ; 268 CHAPTER 5. CHAPTER 5: EXPONENTS AND POLYNOMIALS 4. Perform all additions and subtractions as you come to them, moving left-to-right. 5.1.2.11 Sample Set D Use the order of operations to simplify each of the following. Example 5.22 2 +5=4+5=9 Example 5.23 5 + 3 + 10 = 25 + 9 + 10 = 44 Example 5.24 2 2 2 22 + (5) (8) − 1 = 4 + (5) (8) − 1 = 4 + 40 − 1 = 43 Example 5.25 7 · 6 − 42 + 15 = 7 · 6 − 16 + 1 = 42 − 16 + 1 = 27 Example 5.26 3 (2 + 3) + 72 − 3(4 + 1) 2 3 = (5) + 72 − 3(5) 2 = 125 + 49 − 3 (25) = 125 + 49 − 75 Example 5.27 h i 4(6 + 2) 3 2 = 99 h i2 3 = 4(8) = [4 (512)] 2 2 = [2048] Example 5.28 = 4, 194, 304 6 32 + 22 + 42 = 6 (9 + 4) + 42 = 6 (13) + 42 = 6 (13) + 16 = 78 + 16 Example 5.29 62 +22 42 +6·22 + = 94 13 +82 102 −(19)(5) = = = 36+4 1+64 16+6·4 + 100−95 36+4 1+64 16+24 + 100−95 40 65 40 + 5 = 1 + 13 = 14 Available for free at Connexions < ; 269 5.1.2.12 Practice Set D Exercise 5.1.2.8 3 +4·5 Exercise 5.1.2.9 2 +3 −8·4 Exercise 5.1.2.10  1 + 2 +4 ÷2 Exercise 5.1.2.11   6 10 − 2 − 10 − 6 Exercise 5.1.2.12 Use the order of operations to simplify the following. (Solution on p. 314.) 2 3 4 (Solution on p. 314.) 3 2 2 3 52 +62 −10 1+42 + (Solution on p. 314.) 3 2 2 (Solution on p. 314.) 2 (Solution on p. 314.) 04 −05 72 −6·23 Available for free at Connexions < ; 270 CHAPTER 5. CHAPTER 5: EXPONENTS AND POLYNOMIALS 5.1.2.13 Exercises Exercise 5.1.2.13 b Exercise 5.1.2.14 a Exercise 5.1.2.15 x Exercise 5.1.2.16 (−3) Exercise 5.1.2.17 s Exercise 5.1.2.18 y Exercise 5.1.2.19 a (b + 7) Exercise 5.1.2.20 (21 − x) (x + 5) Exercise 5.1.2.21 xxxxx Exercise 5.1.2.22 (8) (8) xxxx Exercise 5.1.2.23 2 · 3 · 3 · 3 · 3xxyyyyy Exercise 5.1.2.24 2 · 2 · 5 · 6 · 6 · 6xyyzzzwwww Exercise 5.1.2.25 7xx (a + 8) (a + 8) Exercise 5.1.2.26 10xyy (c + 5) (c + 5) (c + 5) Exercise 5.1.2.27 4x4x4x4x4x Exercise 5.1.2.28 (9a) (9a) (9a) (9a) Exercise 5.1.2.29 (−7) (−7) (−7) aabbba (−7) baab Exercise 5.1.2.30 (a − 10) (a − 10) (a + 10) Exercise 5.1.2.31 (z + w) (z + w) (z + w) (z − w) (z − w) Exercise 5.1.2.32 (2y) (2y) 2y2y Exercise 5.1.2.33 For the following problems, write each of the quantities using exponential notation. (Solution on p. 314.) to the fourth squared (Solution on p. 314.) to the eighth cubed 5 times (Solution on p. 314.) squared 3 squared times to the fth cubed minus cubed plus (Solution on p. 314.) squared to the seventh (Solution on p. 314.) (Solution on p. 314.) (Solution on p. 314.) (Solution on p. 314.) (Solution on p. 314.) (Solution on p. 314.) (Solution on p. 314.) 3xyxxy − (x + 1) (x + 1) (x + 1) For the following problems, expand the quantities so that no exponents appear. Available for free at Connexions < ; 271 Exercise 5.1.2.34 4 Exercise 5.1.2.35 6 Exercise 5.1.2.36 7 y Exercise 5.1.2.37 8x y Exercise 5.1.2.38 18x y Exercise 5.1.2.39 9a b Exercise 5.1.2.40 5x 2y Exercise 5.1.2.41 10a b (3c) Exercise 5.1.2.42 (a + 10) a + 10 Exercise 5.1.2.43 3 (Solution on p. 314.) 2 3 2 (Solution on p. 315.) 3 2 2 4 2 (Solution on p. 315.) 3 2 3 3 3 2 (Solution on p. 315.) 2 3 2 2 x2 − y 2 2 2 (Solution on p. 315.) x2 + y 2 Exercise 5.1.2.44 (5x) Exercise 5.1.2.45 (7x) Exercise 5.1.2.46 (a + b) Exercise 5.1.2.47 Exercise 5.1.2.48 For the following problems, select a number (or numbers) to show that 2 2 is not generally equal to 5x2 . is not generally equal to 7x2 . 2 (Solution on p. 315.) is not generally equal to For what real number is For what real numbers, Exercise 5.1.2.49 3 +7 Exercise 5.1.2.50 4 − 18 Exercise 5.1.2.51 5 + 2 (40) Exercise 5.1.2.52 8 + 3 + 5 (2 + 7) Exercise 5.1.2.53 2 + 3 (8 + 1) Exercise 5.1.2.54 2 (6a) a and a2 + b2 . (Solution on p. 315.) equal to b, is 6a2 ? 2 (a + b) equal to a2 + b2 ? Use the order of operations to simplify the quantities for the following problems. (Solution on p. 315.) 2 3 (Solution on p. 315.) 2 2 (Solution on p. 315.) 5 34 + 24 (1 + 5) 3 Available for free at Connexions < ; 272 CHAPTER 5. CHAPTER 5: EXPONENTS AND POLYNOMIALS Exercise 5.1.2.55 6 −4 ÷5 Exercise 5.1.2.56  2 10 − 2 Exercise 5.1.2.57 3 − 4 ÷ 17 Exercise 5.1.2.58 (4 + 3) + 1 ÷ (2 · 5) Exercise 5.1.2.59 2 +2 −2 ·5 ÷4 Exercise 5.1.2.60 1 + 0 + 5 (2 + 8) Exercise 5.1.2.61  (7) (16) − 9 + 4 1 + 3 Exercise 5.1.2.62 Exercise 5.1.2.63 Exercise 5.1.2.64 + Exercise 5.1...
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