B.1. I write a 1, a 2, a Definition 1 Base and Exponent Given a real number a and a natural number n, we define a n by a n a # a # a p a

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1 APPENDIX B B. Review of Integer Exponents B. Review of nth Roots B.3 Review of Rational Exponents B.4 Review of Factoring B.5 Review of Fractional Expressions B.6 Properties of the Real Numbers B. REVIEW OF INTEGER EXPONENTS I write a, a, a 3, etc., for etc. Isaac Newton (June 3, 676) a, aa, aaa, In basic algebra you learned the exponential notation a n, where a is a real number and n is a natural number. Because that definition is basic to all that follows in this and the next two sections of this appendix, we repeat it here. Definition Base and Exponent Given a real number a and a natural number n, we define a n by a n a # a # a p a n factors In the expression a n the number a is the base, and n is the exponent or power to which the base is raised. EXAMPLE Using Algebraic Notation Rewrite each expression using algebraic notation: (a) x to the fourth power, plus five; the fourth power of the quantity x plus five; (c) x plus y, to the fourth power; (d) x plus y to the fourth power. (a) x 4 5 (x 5) 4 (c) (x y) 4 (d) x y 4 Caution: Note that parts (c) and (d) of Example differ only in the use of the comma. So for spoken purposes the idea in (c) would be more clearly conveyed by saying the fourth power of the quantity x plus y. In the case of (d), if you read it aloud to another student or your instructor, chances are you ll be asked, Do you mean x y 4 or do you mean (x y) 4? Moral: Algebra is a precise language; use it carefully. Learn to ask yourself whether what you ve written will be interpreted in the manner you intended. B-

2 B- Appendix B In basic algebra the four properties in the following box are developed for working with exponents that are natural numbers. Each of these properties is a direct consequence of the definition of a n. For instance, according to the first property, we have a a 3 a 5. To verify that this is indeed correct, we note that a a 3 (aa)(aaa) a 5 PROPERTY SUMMARY Property Properties of Exponents Examples. a m a n a mn a 5 a 6 a ; (x )(x ) (x ) 3. (a m ) n a mn ( 3 ) 4 ; [(x ) ] 3 (x ) 6 a mn if m n 3. a m a 6 a a 4 a5 ; a n d a nm if m n a a 6 a4; a 5 if m n a m b m 4. (ab) m a m b m ; a a m (x ) 3 3 (x ) 3 8x 6 ; b b # a x y 3 b 4 x 8 y Of course, we assume that all of these expressions make sense; in particular, we assume that no denominator of a fraction is zero. Now we want to extend our definition of a n to allow for exponents that are integers but not necessarily natural numbers. We begin by defining a 0. Definition Zero Exponent For any nonzero real number a, a 0 (0 0 is not defined.) EXAMPLES (a) 0 (p) (c) a a b b It s easy to see the motivation for defining a 0 to be. Assuming that the exponent zero is to have the same properties as exponents that are natural numbers, we can write a 0 a n a 0n That is, a 0 a n a n Now we divide both sides of this last equation by a n to obtain a 0, which agrees with our definition. Our next definition (see the box that follows) assigns a meaning to the expression a n when n is a natural number. Again, it s easy to see the motivation for this definition. We would like to have a n a n a n(n) a 0

3 B. Review of Integer Exponents B-3 That is, a n a n Now we divide both sides of this last equation by a n to obtain a n a n, in agreement with Definition 3. Definition 3 Negative Exponent For any nonzero real number a and natural number n, a n a n EXAMPLES (a) a 0 0 b 0 (c) x x (d) (a b) 3 (a b) 3 a 6 b 3 (e) It can be shown that the four properties of exponents that we listed earlier continue to hold now for all integer exponents. We make use of this fact in the next three examples. EXAMPLE Using Properties of Exponents Simplify the following expression. Write the answer in such a way that only positive exponents appear. (a b 3 ) (a 5 b) First Method (a b 3 ) (a 5 b) (a b 3 ) # a 5 b (a b 3 ) (a 5 b) a4 b 6 a 5 b b6 b5 54 a a Alternative Method (a b 3 ) (a 5 b) (a 4 b 6 )(a 5 b ) (a b 3 ) (a 5 b) a 45 b 6 a b 5 b5 a as obtained previously EXAMPLE 3 Using Properties of Exponents Simplify the following expression, writing the answer so that negative exponents are not used. a a5 b c 0 3 a 3 b b

4 B-4 Appendix B We show two solutions. The first makes immediate use of the property (a m ) n a mn. In the second solution we begin by working within the parentheses. First Solution Alternative Solution a a5 b c 0 3 a 3 b b a5 b 6 a 9 b 3 a a5 b c 0 a 3 b b 3 a b3 a 8 b 3 ( a 5 b c 0 ) 3 b6(3) b9 9(5) a a 4 ( a 5 b c 0 ) 3 b9 a 4 EXAMPLE EXAMPLE 4 Simplifying with Variable Exponents Simplify the following expressions, writing the answers so that negative exponents are not used. (Assume that p and q are natural numbers.) (a) (a) a 4pq a pq (a p b q ) (a 3p b q ) a 4pq pq a4pq(pq) a a 4pqpq a 3pq (a p b q ) (a 3p b q ) (a p b q )(a 3p b q ) a p b 3q b3q a p 5 Simplifying Numerical Quantities Involving Exponents Use the properties of exponents to compute the quantity 0 # # 6 0 # 3 3 (3 3 )(3 # ) # # 3 3 # 3 # # (3 53 )( 0 ) 3 # 36 As an application of some of the ideas in this section, we briefly discuss scientific notation, which is a convenient form for writing very large or very small numbers. Such numbers occur often in the sciences. For instance, the speed of light in a vacuum is As written, this number would be awkward to work with in calculating. In fact, the number as written cannot even be displayed on some hand-held calculators, because there are too many digits. To write the number 9,979,000,000 (or any positive number) in scientific notation, we express it as a number between and 0, multiplied by an appropriate power of 0. That is, we write it in the form where a 0 and n is an integer. 9,979,000,000 cm/sec a 0 n 0 # # 6.

5 B. Review of Integer Exponents B-5 According to this convention, the number is in scientific notation, but the same quantity written as is not in scientific notation. To convert a given number into scientific notation, we ll rely on the following two-step procedure. To Express a Number Using Scientific Notation. First move the decimal point until it is to the immediate right of the first nonzero digit.. Then multiply by 0 n or 0 n, depending on whether the decimal point was moved n places to the left or to the right, respectively. For example, to express the number 9,979,000,000 in scientific notation, first we move the decimal point 0 places to the left so that it s located between the and the 9, then we multiply by 0 0. The result is or, more simply, 9,979,000, ,979,000, As additional examples we list the following numbers expressed in both ordinary and scientific notation. For practice, you should verify each conversion for yourself using our two-step procedure. 55, All scientific calculators have keys for entering numbers in scientific notation. (Indeed, many calculators will convert numbers into scientific notation for you.) If you have questions about how your own calculator operates with respect to scientific notation, you should consult the user s manual. We ll indicate only one example here, using a Texas Instruments graphing calculator. The keystrokes on other brands are quite similar. Consider the following number expressed in scientific notation: (This gargantuan number is the diameter, in miles, of a galaxy at the center of a distant star cluster known as Abell 09.) The sequence of keystrokes for entering this number on the Texas Instruments calculator is 3.59 EE 0 EXERCISE SET B. A In Exercises 4, evaluate each expression using the given value of x.. x 3 x 4; x. x x 3x 3 ; x x 3. (x ) x 4. x x 3; (x ) ; In Exercises 5 6, use the properties of exponents to simplify each expression. 5. (a) a 3 a 6. (a) (3 ) 3 ( 3 ) (a ) 3 (a ) (x ) 3 (x 3 ) (c) (a ) (a ) 3 (c) (x ) a (x a ) 7. (a) yy y 8 (y )(y ) (y ) 8 (c) [(y )(y ) 8 ]

6 B-6 Appendix B 8. (a) x (x 3) 0 9. (a) x 0 (x 3) 9 x 0 (x 3) 9 x (x 3) 0 (c) 0 0 (c) 9 0. (a) (y y 3 ). (a) y (y 3 ) t 5 t 9 t 9 (c) m ( n ) (c) (t 3) 5 (t 3) 9. (a) x 6 x 6 y 5 3. (a) x x y 0 x x y 0 x 6 x 6 y 5 (c) (x 3x ) 6 x 3x (c) a x y 0 x 6 y b 5 4. (a) (x y 3 z) 4 5. (a) 4(x 3 ) (x y 3 z) 4 (4x 3 ) (c) (x y 3 z) 4 (4x ) 3 (c) (4x 3 ) 6. (a) (x ) 7 (x ) 7 [(x )] 7 (x ) 7 (c) (x ) 7 [(x )] 7 For Exercises 7 38, simplify each expression. Write the answers in such a way that negative exponents do not appear. In Exercises 35 38, assume that the letters p and q represent natural numbers. 7. (a) (a) (64 3 ) 0 ( ) 3 (c) (64 0 ) 3 (c) ( ) 0 9. (a) (a) 4 4 (0 0 ) [ 4 4 ] (c) [(0 )(0 )] (c) [ 4 4 ] (a) (a) (5 0) (a bc 0 ) 3 6. (a 3 b) 3 (a b 4 ) 7. (a b c 3 ) 8. ( 0 ) 9. a x3 y z 3 xy z b a x4 y 8 z xy z b a a3 b 9 c 0 a x4 y 8 z xy a 5 b c b z b 6 4 t 5 x (x ) 3 (x ) 3 y y 3 3 x b p b p (b ) q 37. (x p )(x p ) 38. ( pq )( qp ) In Exercises 39 4, use the properties of exponents in computing each quantity, as in Example 5. (The point here is to do as little arithmetic as possible.) # # # 3 0 # # 4 For Exercises 43 50, express each number in scientific notation. 43. The average distance (in miles) from Earth to the Sun: 44. The average distance (in miles) from the planet Pluto to the Sun: 45. The average orbital speed (in miles per hour) of Earth: 46. The average orbital speed (in miles per hour) of the planet Mercury: 47. The average distance (in miles) from the Sun to the nearest star: 48. The equatorial diameter (in miles) of (a) Mercury: 303 (c) Jupiter: 88,733 Earth: 797 (d) the Sun: 865, The time (in seconds) for light to travel (a) one foot: across an atom: (c) across the nucleus of an atom: 50. The mass (in grams) of (a) a proton: an electron: 9,900,000 3,666,000,000 66,800 07,300 b pq 5,000,000,000,000,000, a 44 # 5 b 3 #

7 B. Review of nth Roots B-7 B. Hindu mathematicians first recognized negative roots, and the two square roots of a positive number,... though they were suspicious also. David Wells in The Penguin Dictionary of Curious and Interesting Numbers (Harmondsworth, Middlesex, England: Penguin Books, Ltd., 986) REVIEW OF nth ROOTS In this section we generalize the notion of square root that you studied in elementary algebra. The new idea is that of an nth root; the definition and some basic examples are given in the box that follows. Definition nth Roots Let n be a natural number. If a and b are real numbers and a n b then we say that a is an nth root of b. When n and when n 3, we refer to the roots as square roots and cube roots, respectively. EXAMPLES Both 3 and 3 are square roots of 9 because 3 9 and (3) 9. Both and are fourth roots of 6 because 4 6 and () 4 6. is a cube root of 8 because is a fifth root of 43 because (3) As the examples in the box suggest, square roots, fourth roots, and all even roots of positive numbers occur in pairs, one positive and one negative. In these cases we use the notation n b to denote the positive, or principal, nth root of b. As examples of this notation, we can write (The principal fourth root of 8 is 3.) (The principal fourth root of 8 is not 3.) (The negative of the principal fourth root of 8 is 3.) The symbol is called a radical sign, and the number within the radical sign is the n radicand. The natural number n used in the notation is called the index of the radical. For square roots, as you know from basic algebra, we suppress the index and simply write rather than. So, for example, 5 5. On the other hand, as we saw in the examples, cube roots, fifth roots, and in fact, all odd roots occur singly, not in pairs. In these cases, we again use the notation n b for the nth root. The definition and examples in the following box summarize our discussion up to this point. Definition Principal nth Root. Let n be a natural number. If a and b are nonnegative real numbers, then n b a if and only if b a n The number a is the principal nth root of b.. If a and b are negative and n is an odd natural number, then n b a if and only if b a n EXAMPLES 5 5; 5 5; B6 ; B 3 There are five properties of nth roots that are frequently used in simplifying certain expressions. The first four are similar to the properties of square roots that are developed in elementary algebra. For reference we list these properties side by side

8 B-8 Appendix B in the following box. Property 5 is listed here only for the sake of completeness; we ll postpone discussing it until Appendix B.3. PROPERTY SUMMARY Properties of nth Roots Suppose that x and y are real numbers and that m and n are natural numbers. Then each of the following properties holds, provided only that the expressions on both sides of the equation are defined (and so represent real numbers)... n x n x n n n xy x y CORRESPONDING PROPERTIES FOR SQUARE ROOTS x x xy xy 3. n x n x B y n y x x B y y 4. n even: n x n 0 x 0 n odd: n x n x m 5. n x x mn x 0 x 0 Our immediate use for these properties will be in simplifying expressions involving nth roots. In general, we try to factor the expression under the radical so that one factor is the largest perfect nth power that we can find. Then we apply Property or 3. For instance, the expression 7 is simplified as follows: In this procedure we began by factoring 7 as (36)(). Note that 36 is the largest factor of 7 that is a perfect square. If we were to begin instead with a different factorization, for example, 7 (9)(8), we could still arrive at the same answer, but it would take longer. (Check this for yourself.) As another example, let s simplify First, what (if any) is the largest perfect-cube factor of 40? The first few perfect cubes are so we see that 8 is a perfect-cube factor of 40, and we write (8)(5) EXAMPLE Simplifying Square Roots 7 (36)() Simplify: (a) 75; B 49. (a) 75 (4)(3) (5)(3) B ( 5)

9 B. Review of nth Roots B-9 EXAMPLE Simplifying nth Roots Simplify: EXAMPLE 3 Simplifying Radicals Containing Variables Simplify each of the following expressions by removing the largest possible perfectsquare or perfect-cube factor from within the radical: (a) 8x ; 8x, where x 0; (c) 8a 7, where a 0; (d) 3 6y 5. (a) 8x (4)()(x ) 4x 0 x 0 8x 4x (c) x x x because x 0 8a 7 (9a 6 )(a) 9a 6 a 3a 3 a (d) 3 6y 5 3 8y 3 3 y y 3 y EXAMPLE 4 Simplifying nth Roots Containing Variables Simplify each of the following, assuming that a, b, and c are positive: (a) 8ab c 5 ; 4 3a 6 b 5. B c 8 (a) 8ab c 5 (4b c 4 )(ac) 4b c 4 ac 4 3a 6 b 5 4 3a 6 b 5 B c 8 4 c 8 4 6a 4 b 4 4 a b c bc ac ab4 a b c

10 B-0 Appendix B In Examples through 4 we used the definitions and properties of nth roots to simplify certain expressions. The box that follows shows some common errors to avoid in working with roots. Use the error box to test yourself: Cover up the columns labeled Correction and Comment, and try to decide for yourself where the errors lie. Errors to Avoid Error Correction Comment Although is one of the fourth roots of 6, it is not the principal fourth root. The notation 4 6 is reserved for the principal fourth root Although 5 is one of the square roots of 5, it is not the principal square root. a b a b The expression The properties of roots differ with a b cannot respect to addition versus multibe simplified. plication. For ab we do have the simplification ab ab (assuming that a and b are nonnegative). 3 a b 3 a 3 b The expression For 3 ab we do have 3 ab 3 a 3 b. 3 a bcannot be simplified. There are times when it is convenient to rewrite fractions involving radicals in alternative forms. Suppose, for example, that we want to rewrite the fraction 53 in an equivalent form that does not involve a radical in the denominator. This is called rationalizing the denominator. The procedure here is to multiply by in this way: 5 53 That is, as required. 3 3, To rationalize a denominator of the form a b, we need to multiply the fraction not by bb, but rather by a ba b ( ). To see why this is necessary, notice that which still contains a radical, whereas # 5 # a bb ab b a ba b a ab ab b a b which is free of radicals. Note that we multiply a b by a b to obtain an expression that is free of square roots by taking advantage of the form of a difference

11 B. Review of nth Roots B- of two squares. Similarly, to rationalize a denominator of the form a b, we multiply the fraction by a ba b. (The quantities a b and a b are said to be conjugates of each other.) The next two examples make use of these ideas. EXAMPLE 5 Rationalizing a Square Root Denominator Simplify: 350. First, we rationalize the denominator in the fraction : Next, we simplify the expression 350: 350 3(5)() 35 (3)(5) 5 Now, putting things together, we have # # ( 30) EXAMPLE 6 Using the Conjugate to Rationalize a Denominator Rationalize the denominator in the expression 4 3. We multiply by, writing it as # 4 3 # Note: Check for yourself that multiplying the original fraction by eliminate radicals in the denominator. 3 does not In the next example we are asked to rationalize the numerator rather than the denominator. This is useful at times in calculus.

12 B- Appendix B EXAMPLE 7 Rationalizing a Numerator Using the Conjugate x 3 Rationalize the numerator of: where x 0, x 3. x 3, x 3 x 3 # x 3 x 3 # x 3 x 3 x 3 (x 3)x 3 x 3 (x 3)x 3 x 3 The strategy for rationalizing numerators or denominators involving nth roots is similar to that used for square roots. To rationalize a numerator or a denominator involving an nth root, we multiply the numerator or denominator by a factor that yields a product that itself is a perfect nth power. The next example displays two instances of this. EXAMPLE 8 Rationalizing Denominators Containing nth Roots 6 (a) Rationalize the denominator of: 3 7. ab Rationalize the denominator of: where a 0, b 0. 4 a b 3, (a) # 6 # as required ab # ab # 4 a b 4 a b 3 4 a b 3 4 a b ab4 a b ab 4 a b 4 a b 4 a 4 b 4 ab Finally, just as we used the difference of squares formula to motivate the method of rationalizing certain two-term denominators containing a square root term, we can use the difference or sum of cubes formula (see Appendix B.4) to rationalize a denominator involving a cube root term. In particular, in the sum of cubes formula x 3 y 3 (x y)(x xy y ), the expression (x xy y ) is the conjugate of the expression x y.

13 B. Review of nth Roots B-3 EXAMPLE 9 Using the Conjugate to Rationalize a Denominator Involving a Cube Root Rationalize the denominator in the expression 3 a 3 b. 3 a If we let x and y the conjugate of the denominator x y 3 a 3 b 3 b, is x xy y 3 a 3 a 3 b 3 b 3 a 3 ab 3 b where, for example, 3 a 3 a 3 a 3 a. Then 3 a 3 b 3 a 3 b # 3 a 3 ab 3 b 3 a 3 ab 3 b 3 a 3 ab 3 b 3 a 3 ab 3 b 3 a 3 3 b 3 a b EXERCISE SET B. A In Exercises 8, determine whether each statement is TRUE or FALSE (5) 5 For Exercises 9 8, evaluate each expression. If the expression is undefined (i.e., does not represent a real number), say so. 9. (a) (a) (a) (a) (a) 6 4. (a) (a) (a) (a) (a) 4 (0) (0) 3 In Exercises 9 44, simplify each expression. Unless otherwise specified, assume that all letters in Exercises represent positive numbers. 9. (a) 8 0. (a) (a) 98. (a) (a) (a) (a) 8 6. (a) (a) 8. (a) 9. (a) 30. (a) (a) 36x 36. (a) 5x 4 y 3 36y, where y 0 4 6a 4, where a (a) ab a b 38. (a) 3 5x 6 ab 3 a 3 b 4 64y 4, where y a 3 b 4 c (a b) 5 (6a b ) a 4 b a 4 b a b c ab 3 4 a 3 b

14 B-4 Appendix B In Exercises 45 70, rationalize the denominators and simplify where possible. (Assume that all letters represent positive quantities.) a b a b a 4 b a 5 b 6. (a) 3 6. (a) (a) x x 64. (a) x 5 x x x y x a x 65. (a) 66. (a) x x x a x a 3 y 3 y x y x y 67. x h x 68. x h x h a a 3 b In Exercises 7 76, rationalize the numerator. 7. x 5(x 5) 7. y 3(4y 9) 73. h h 74. h (h) 75. x h xh 76. x h x hh In Exercises 77 80, refer to the following table. The left-hand column of the table lists four errors to avoid in working with expressions containing radicals. In each case, give a numerical example showing that the expressions on each side of the equation are, in general, not equal. Use a calculator as necessary. Numerical Example Showing that the Formula Is Not, Errors to Avoid in General, Valid 77. a b a b 78. x y x y u v 3 u 3 v p 3 q 3 p q B 8. (a) Use a calculator to evaluate 8 7 and 7. What do you observe? Prove that Hint: In view of the definition of a principal square root, you need to check that Use a calculator to provide empirical evidence indicating that both of the following equations may be correct: () p The point of this exercise is to remind you that as useful as calculators may be, there is still the need for proofs in mathematics. In fact, it can be shown that the first equation is indeed correct, but the second is not. If Descartes... had discarded the radical sign altogether and had introduced the notation for fractional as well as integral exponents,... it is conceivable that generations upon generations of pupils would have been saved the necessity of mastering the operations with two... notations when one alone (the exponential) would have answered all purposes. Florian Cajori, A History of Mathematical Notations, Vol. (La Salle, Ill.: The Open Court Publishing Co., 98) REVIEW OF RATIONAL EXPONENTS... Nicole Oresme, a bishop in Normandy (about 33 38), first conceived the notion of fractional powers, afterwards rediscovered by [Simon] Stevin [548 60]... Florian Cajori in A History of Elementary Mathematics (New York: Macmillan Co., 97) We can use the concept of an nth root to give a meaning to fractional exponents that is useful and, at the same time, consistent with our earlier work. First, by way of motivation, suppose that we want to assign a value to 5 3. Assuming that the usual properties of exponents continue to apply here, we can write That is, or B.3 (5 3 ) (5 3 )

15 B.3 Review of Rational Exponents B-5 By replacing 5 and 3 with b and n, respectively, we can see that we want to define b n to mean n b. Also, by thinking of b mn as (b n ) m, we see that the definition for b mn ought to be n b m. These definitions are formalized in the box that follows. Definition Rational Exponents. Let b denote a real number and n a natural number. We define b n by (If n is even, we require that b 0.). Let mn be a rational number reduced to lowest terms. Assume that n is positive and that n b exists. Then or, equivalently, b n n b b mn n b m b mn n b m EXAMPLES 4 4 (8) or, equivalently, It can be shown that the four properties of exponents that we listed in Appendix B. (on page B-) continue to hold for rational exponents in general. In fact, we ll take this for granted rather than follow the lengthy argument needed for its verification. We will also assume that these properties apply to irrational exponents. For instance, we have (The definition of irrational exponents is discussed in Section 5..) In the next three examples we display the basic techniques for working with rational exponents. EXAMPLE Evaluating Rational Exponents Simplify each of the following quantities. Express the answers using positive exponents. If an expression does not represent a real number, say so. (a) (c) (49) (d) 49 (a) (49 ) 49 7 (c) The quantity (49) does not represent a real number because there is no real number x such that x 49. (d) Alternatively, we have 49 (49 ) 7 7

16 B-6 Appendix B EXAMPLE Simplifying Expressions Containing Rational Exponents Simplify each of the following. Write the answers using positive exponents. (Assume that a 0.) 5 6a 3 (a) (5a 3 )(4a 34 ) (c) (x ) 5 (x ) 45 B a 4 (a) (5a 3 )(4a 34 ) 0a (3)(34) 0a 7 because a 3 a 6a3 5 b B a 4 a 4 (6a ) 5 because a 60 (c) (x ) 5 (x ) 45 (x ) x EXAMPLE 3 Evaluating Rational Exponents Simplify: (a) 3 5 ; (8) 43. (a) Alternatively, we have 3 5 ( 5 ) (8) () 4 6 Alternatively, we can write (8) 43 [() 3 ] 43 () 4 6 Rational exponents can be used to simplify certain expressions containing radicals. For example, one of the properties of nth roots that we listed but did not discuss m in Appendix B. is n x x. mn Using exponents, it is easy to verify this property. We have m n x (x n ) m x mm mn x as we wished to show EXAMPLE 4 Using Rational Exponents to Combine Radicals Consider the expression x 3 y, where x and y are positive. (a) Rewrite the expression using rational exponents. Rewrite the expression using only one radical sign. (a) x 3 y x y 3 x 3 y x y 3 x 36 y 46 6 x 3 6 y 4 6 x 3 y 4 rewriting the fractions using a common denominator

17 B.3 Review of Rational Exponents B-7 EXAMPLE 5 Using Rational Exponents to Simplify a Radical Expression Rewrite the following expression using rational exponents (assume that x, y, and z are positive): C 3 x 4 y 3 5 z 4 C 3 x 4 y 3 a 3 x 4 y 3 b a x3 y 34 b 5 z 4 5 z 4 z 45 x6 y 38 z 5 x 6 y 38 z 5 EXERCISE SET B.3 A 5 Exercises 4 are warm-up exercises to help you become familiar with the definition b mn b m b m. In Exercises 8, write the expression in the two equivalent forms b m and n b m. In Exercises 9 4, write the expression in the form b mn.. a 35. x (x ) (a b) xy3 8. (a)b 9. p 0. 5 R 3. 7 ( u) p 3p (a b ) 3 4. (a b ) p ( u) In Exercises 5 50, evaluate or simplify each expression. Express the answers using positive exponents. If an expression is undefined (that is, does not represent a real number), say so. Assume that all letters represent positive numbers (36) (6) 0. () (8) (3) 5 8. (5) 3 9. (000) (43) (49) 34. (64) (0.00) () (96) 5 (0007) (56 34 ) (a 3 )(3a 4 ) 46. 3a a 3 a (x ) 3 (x ) 43 (x ) (x ) (x ) 65 (x ) 65 (x ) 5 (x ) 95 For Exercises 5 58, follow Example 4 in the text to rewrite each expression in two ways: (a) using rational exponents and using only one radical sign. (Assume that x, y, and z are positive.) x 5 y x 3 y 4 z 57. x a x b x a x x x In Exercises 59 66, rewrite each expression using rational exponents rather than radicals. Assume that x, y, and z are positive (x ) x y 6. x x 3 x x 4 y x 3 y 4 z 67. Use a calculator to determine which is larger, 9 09 or Use a calculator to determine which number is closer to 3: 3 37 or In Exercises 69 74, refer to the following table. The left-hand column of the table lists six errors to avoid in working with expressions containing fractional exponents. In each case, give a numerical example showing that the expressions on each side of the equation are not equal. Use a calculator as necessary. Numerical Example Showing That the Formula Is Not, Errors to Avoid in General, Valid 69. (a b) a b 70. (x y ) x y 7. (u v) 3 u 3 v 3 7. (p 3 q 3 ) 3 p q 73. x m x m 74. x x

18 B-8 Appendix B fac # tor (fakt 0 r), n..... Math. one of two or more numbers, algebraic expressions, or the like, that when multiplied together produce a given product;... v.t. 0. Math. to express (a mathematical quantity) as a product of two or more quantities of like kind, as 30 # 3 # 5, or x y (x y)(x y). The Random House Dictionary of the English Language, nd ed. (New York: Random House, 987) B.4 REVIEW OF FACTORING There are many cases in algebra in which the process of factoring simplifies the work at hand. To factor a polynomial means to write it as a product of two or more nonconstant polynomials. For instance, a factorization of x 9 is given by x 9 (x 3)(x 3) In this case, x 3 and x 3 are the factors of x 9. There is one convention that we need to agree on at the outset. If the polynomial or expression that we wish to factor contains only integer coefficients, then the factors (if any) should involve only integer coefficients. For example, according to this convention, we will not consider the following type of factorization in this section: x x x because it involves coefficients that are irrational numbers. (We should point out, however, that factorizations such as this are useful at times, particularly in calculus.) As it happens, x is an example of a polynomial that cannot be factored using integer coefficients. In such a case we say that the polynomial is irreducible over the integers. If the coefficients of a polynomial are rational numbers, then we do allow factors with rational coefficients. For instance, the factorization of y numbers is given by y 4 a y bay b over the rational We ll consider five techniques for factoring in this section. These techniques will be applied in the next two sections and throughout the text. In Table we summarize these techniques. Notice that three of the formulas in the table are just restatements 4 TABLE Basic Factoring Techniques Technique Example or Formula Remark Common factor 3x 4 6x 3 x 3x (x x 4) In any factoring problem, the first step always is 4(x ) x(x ) (x )(4 x) to look for the common factor of highest degree. Difference of x a (x a)(x a) There is no corresponding formula for a sum of squares squares; x a is irreducible over the integers. Trial and error x x 3 (x 3)(x ) In this example the only possibilities, or trials, are (a) (x 3)(x ) (c) (x 3)(x ) (x 3)(x ) (d) (x 3)(x ) By inspection or by carrying out the indicated multiplications, we find that only case (c) checks. Difference of cubes x 3 a 3 (x a)(x ax a ) Verify these formulas for yourself by carrying out Sum of cubes x 3 a 3 (x a) (x ax a ) the multiplications. Then memorize the formulas. Grouping x 3 x x (x 3 x ) (x ) This is actually an application of the common x (x ) (x ) # factor technique. (x )(x )

19 B.4 Review of Factoring B-9 of Special Products formulas. Remember that it is the form or pattern in the formula that is important, not the specific choice of letters. The idea in factoring is to use one or more of these techniques until each of the factors obtained is irreducible. The examples that follow show how this works in practice. EXAMPLE Factoring Using the Difference of Squares Technique Factor: (a) x 49; (a 3b) 49. (a) x 49 x 7 (x 7)(x 7) difference of squares Notice that the pattern or form is the same as in part (a): It s a difference of squares. We have (a 3b) 49 (a 3b) 7 [(a 3b) 7][(a 3b) 7] (a 3b 7)(a 3b 7) difference of squares EXAMPLE Factoring Using Common Factor and Difference of Squares Factor: (a) x 3 50x; 3x 5 3x. (a) x 3 50x x(x 5) common factor x(x 5)(x 5) difference of squares 3x 5 3x 3x(x 4 ) common factor 3x[(x ) ] 3x(x )(x ) difference of squares 3x(x )(x )(x ) difference of squares, again EXAMPLE 3 Factoring by Trial and Error Factor: (a) x 4x 5; (a b) 4(a b) 5. (a) x 4x 5 (x 5)(x ) trial and error Note that the form of this expression is ( ) 4( ) 5 This is the same form as the expression in part (a), so we need only replace x with the quantity a b in the solution for part (a). This yields EXAMPLE 4 Factoring by Trial and Error Factor: z 4 9z 4. (a b) 4(a b) 5 [(a b) 5][(a b) ] (a b 5)(a b ) We use trial and error to look for a factorization of the form (z?)(z?). There are three possibilities: (i) (z )(z ) (ii) (z )(z 4) (iii) (z 4)(z )

20 B-0 Appendix B Each of these yields the appropriate first term and last term, but (after checking) only possibility (ii) yields 9z for the middle term. The required factorization is then z 4 9z 4 (z )(z 4). Question: Why didn t we consider any possibilities with subtraction signs in place of addition signs? EXAMPLE 5 Two Irreducible Expressions Factor: (a) x 9; x x 3. (a) The expression x 9 is irreducible over the integers. (This can be discovered by trial and error.) If the given expression had instead been x 9, then it could have been factored as a difference of squares. Sums of squares, however, cannot in general be factored over the integers. The expression x x 3 is irreducible over the integers. (Check this for yourself by trial and error.) EXAMPLE 6 Factoring Using Grouping and a Special Product Factor: x y 0x 5. Familiarity with the special product (a b) a ab b suggests that we try grouping the terms this way: (x 0x 5) y Then we have x y 0x 5 (x 0x 5) y (x 5) y [(x 5) y][(x 5) y] (x y 5)(x y 5) difference of squares EXAMPLE 7 Factoring Using Grouping Factor: ax ay bx by. We factor a from the first two terms and b from the second two to obtain ax ay bx by a(x y ) b(x y ) We now recognize the quantity x y as a common expression that can be factored out. We then have a(x y ) b(x y ) (x y )(a b) The required factorization is therefore Alternatively, ax ay bx by (x y )(a b) ax ay bx by (ax bx) (ay by ) x(a b) y (a b) (a b)(x y )

21 B.4 Review of Factoring B- EXAMPLE 8 Factoring Sum and Difference of Cubes Factor: (a) t 3 5; 8 (a ) 3. (a) t 3 5 t (t 5)(t 5t 5) 8 (a ) 3 3 (a ) 3 difference of cubes [ (a )][ (a ) (a ) ] a(4 a 4 a 4a 4) a(a 6a ) sum of cubes As you can check now, the expression a 6a is irreducible over the integers. Therefore the required factorization is a(a 6a ). For some calculations (particularly in calculus) it s helpful to be able to factor an expression involving fractional exponents. For instance, suppose that we want to factor the expression x(x ) (x ) 3 The common expression to factor out here is (x ) ; the technique is to choose the expression with the smaller exponent. EXAMPLE 9 Factoring with Rational Exponents Factor: x(x ) (x ) 3. x(x ) (x ) 3 (x ) [x (x ) 3 ] (x ) [x (x ) ] (x ) (4x 3x ) We conclude this section with examples of four common errors to avoid in factoring. The first two errors in the box that follows are easy to detect; simply multiplying out the supposed factorizations shows that they do not check. The third error may result from a lack of familiarity with the basic factoring techniques listed on page B-8. The fourth error indicates a misunderstanding of what is required in factoring; the final quantity in a factorization must be expressed as a product, not a sum, of terms or expressions. Errors to Avoid Error Correction Comment x 6x 9 (x 3) x 6x 9 (x 3) Check the middle term. x 64 (x 8)(x 8) x 64 is irreducible A sum of squares is, in general, irreducible over over the integers. the integers. (A difference of squares, however, can always be factored.) x 3 64 is irreducible x 3 64 (x 4)(x 4x 6) Although a sum of squares is irreducible, a sum over the integers of cubes can be factored. x x 3 factors as x x 3 is irreducible over The polynomial x x 3isthesum of the expresx(x ) 3 the integers. sion x(x ) and the constant 3. By definition, however, the factored form of a polynomial must be a product (of two or more nonconstant polynomials).

22 B- Appendix B EXERCISE SET B.4 A In Exercises 66, factor each polynomial or expression. If a polynomial is irreducible, state this. (In Exercises 6 the factoring techniques are specified.). (Common factor and difference of squares) (a) x 64 (c) z z 3 7x 4 4x (d) a b c. (Common factor and difference of squares) (a) t 4 (c) u v 5 x 6 x 5 x 4 (d) 8x 4 x 3. (Trial and error) (a) x x 3 (c) x x 3 x x 3 (d) x x 3 4. (Trial and error) (a) x 7x 4 (c) x 7x 4 x 7x 4 (d) x 7x 4 5. (Sum and difference of cubes) (a) x 3 (c) 000 8x 6 x 3 6 (d) 64a 3 x (Grouping) (a) x 4 x 3 3x 6 a x bx a z bz 7. (a) 44 x 8. (a) 4a b 9c 44 x 4a b 9c (c) 44 (y 3) (c) 4a b 9(ab c) 9. (a) h 3 h 5 00h 3 h 5 (c) 00(h ) 3 (h ) 5 0. (a) x 4 x 3x 4 48x (c) 3(x h) 4 48(x h). (a) x 3x 40. (a) x 0x 6 x 3x 40 x 0x 6 3. (a) x 5x (a) x x 6 x 3x 36 x x 6 5. (a) 3x x 6 6. (a) x 4x 3x x 6 x 4x 3 7. (a) 6x 3x 5 8. (a) 6x 8x 9 6x x 5 6x 43x 9 9. (a) t 4 t 0. (a) t 4 9t 0 t 4 t t 4 9t 0 (c) t 4 t (c) t 4 9t 0. (a) 4x 3 0x 5x. (a) x 3 x x 4x 3 0x 5x x 3 x x 3. (a) ab bc a ac (u v)x xy (u v) (u v)y 4. (a) 3(x 5) 3 (x 5) a(x 5) 3 b(x 5) 5. x z xzt xyz yt 6. a t b t cb ca 7. a 4 4a b c 4b 4 c 4 8. A B 6A A B 30. x x (a b) (x y) 3 y (a b) 3 8c x 3 y 3 x y 36. 8a 3 7b 3 a 3b 37. (a) p (a) p 4 4 p 8 p x 3 3x 3x 40. 6x x 8x 3 4. x 6y 4. 4u 5v c 44. y 6 4 (a b) a b z x m 3 n 3 8 x x xy y 50. x x 5. 64(x a) 3 x a 5. 64(x a) 4 x a 53. x a y xy 54. a 4 (b c) x 3 8x 39x 56. x 3 a 8y 3 a 4x 3 b 3y 3 b 57. xy 5 4x 9y 58. ax ( ab)xy by 59. ax (a b)x b 60. (5a a 0) (4a 5a 6) 6. (x ) (x ) 3 6. (x ) 3 (x ) (x ) (x ) (x ) 3 (x ) (x 3) (x 3) (ax b) 3 ax bb In Exercises 67 and 68, evaluate the given expressions using factoring techniques. (The point here is to do as little actual arithmetic as possible.) 67. (a) (a) (c) (c) 5 0 B In Exercises 69 73, factor each expression. 69. A 3 B 3 3AB(A B) 70. p 3 q 3 p(p q ) q(p q) 7. x(a x ) x 3 (a x ) 3 7. (x a) (x a) (x a) (x a) y 4 (p q)y 3 (p q pq )y p q 74. (a) Factor x 4 x y y 4. Factor x 4 x y y 4. Hint: Add and subtract a term. [Keep part (a) in mind.] (c) Factor x 6 y 6 as a difference of squares. (d) Factor x 6 y 6 as a difference of cubes. [Use the result in part to obtain the same answer as in part (c).]

23 B.5 Review of Fractional Expressions B-3 But if you do use a rule involving mechanical calculation, be patient, accurate, and systematically neat in the working. It is well known to mathematical teachers that quite half the failures in algebraic exercises arise from arithmetical inaccuracy and slovenly arrangement. George Chrystal (893) B.5 REVIEW OF FRACTIONAL EXPRESSIONS The rules of basic arithmetic that you learned for working with simple fractions are also used for fractions involving algebraic expressions. In algebra, as in arithmetic, we say that a fraction is reduced to lowest terms or simplified when the numerator and denominator contain no common factors (other than and ). The factoring techniques that we reviewed in Appendix B.4 are used to reduce fractions. For example, to reduce the fraction we write x 9 x 3x, x 9 (x 3)(x 3) x 3 x 3x x(x 3) x In the box that follows, we review two properties of negatives and fractions that will be useful in this section. Notice that Property follows from Property because a b b a a b (a b) PROPERTY SUMMARY Negatives and Fractions Property Example. b a (a b) x (x ). a b x 5 b a 5 x EXAMPLE Using Property to Simplify a Fractional Expression Simplify: 4 3x 5x x 5x 0 4 3x 5(3x 4) 4 3x a using Property 5 3x 4 b 5 In Example we display two more instances in which factoring is used to reduce a fraction. After that, Example 3 indicates how these skills are used to multiply and divide fractional expressions. EXAMPLE Using Factoring to Simplify a Fractional Expression x 3 8 x 6x 8 Simplify: (a) x a(x ) b(x ). x ; (a) x 3 8 x x (x )(x x 4) x x 4 x x(x ) x 6x 8 (x )(x 4) x 4 a(x ) b(x ) (x )(a b) a b

24 B-4 Appendix B EXAMPLE 3 Using Factoring to Simplify Products and Quotients Carry out the indicated operations and simplify: (a) x 3 # x 8 x 3x 4x 6x ; (c) x 44 x 4 x x. x x 6 x x # x3 4 x ; (a) x 3 # x 8 x 3x 4x 6x x 3 # 6(x 3) x(x 3) x(x 3) 6x 3 x (x 3) 3x x 3 x x 6 # x3 (x 3)(x ) x # (x )(x x ) x 4 x (x x ) ( x)( x) (x 3)(x ) using the fact x that x x x 3 x x x 44 x (c) x 4 x x 44 # x x 4 x (x )(x ) (x )(x ) x # x x x As in arithmetic, to combine two fractions by addition or subtraction, the fractions must have a common denominator, that is, the denominators must be the same. The rules in this case are For example, we have 4x x a b c a c b b x x and 4x (x ) x Fractions with unlike denominators are added or subtracted by first converting to a common denominator. For instance, to add 9a and 0a, we write 9 0 a a 9 # a 0 a a a a b c a c b b 9a 0 9a 0 a a a Notice that the common denominator used was a. This is the least common denominator. In fact, other common denominators (such as a 3 or a 4 ) could be used here, but that would be less efficient. In general, the least common denominator for a given group of fractions is chosen as follows. Write down a product involving the irreducible factors from each denominator. The power of each factor should be equal to (but not greater than) the highest power of that factor appearing in any of the individual denominators. For example, the least common denominator for the two fractions 4x x x x x

25 B.5 Review of Fractional Expressions B-5 and is (x ) (x ). In the example that follows, notice (x ) (x )(x ) that the denominators must be in factored form before the least common denominator can be determined. EXAMPLE 4 Using the Least Common Denominator Combine into a single fraction and simplify: 3 x (a) (c) x 9 4x 7x 0y x 3 ; ; (a) Denominators: # x; # 5 # y ; Least common denominator; # 5xy 0xy Note: The final numerator is irreducible over the integers. Denominators: x 9 (x 3)(x 3); x 3 Least common denominator: (x 3)(x 3) (c) 3 4x x x 9 x 3 7x 0y 3 4x # 5y 5y 5 (x 4x 3) (x )(x 3) 5y 4x 40xy 0xy x (x 3)(x 3) 5 x x x 6 x 5 x (x )(x 3) x 5 x # x 3 (x )(x 3) x x 3 (x )(x 6) (x )(x 3) 7x # x 0y x # 0xy 0xy x 3 x (x 3)(x 3) # x 3 x 3 x 3 x (x 3) (x 3)(x 3) 3 (x 3)(x 3) 5 x x x 6 x. Notice in the last line that we were able to simplify the answer by factoring the numerator and reducing the fraction. (This is why we prefer to leave the least common denominator in factored form, rather than multiplying it out, in this type of problem.) The next four examples illustrate using the least common denominator to simplify compound fractions, or complex fractions, which are fractions within fractions. x 4x (x )(x 3) x 6 x 3 using the fact that x (x ) x 6 x 3

26 B-6 Appendix B EXAMPLE 5 Simplifying a Compound Fraction Simplify: 3a 4b 5 6a. b The least common denominator for the four individual denominators 3a,4b,6a, and b is a b. Multiplying the given expression by a ba b, which equals, yields a b # 3a 4b 4ab 3a a b 5 6a 0b a b EXAMPLE 6 Simplifying a Fraction Containing Negative Exponents Simplify: (x y ). (The answer is not x y.) After applying the definition of negative exponent to rewrite the given expression, we ll use the method shown in Example 5. (x y ) a x y b xy # xy x y x y xy y x multiplying by xy xy EXAMPLE 7 Simplifying a Compound Fraction x h x Simplify:. (This type of expression occurs in calculus.) h x h x (x h)x # x h x h (x h)x h x (x h) (x h)xh h (x h)xh x(x h) (x h)x multiplying by (x h)x EXAMPLE 8 Simplifying a Compound Fraction x x Simplify:. x

27 B.5 Review of Fractional Expressions B-7 x x x # x x multiplying by x x x x3 x x 3 x x x (x )(x x ) x x (x )(x ) x EXERCISE SET B.5 A In Exercises, reduce the fractions to lowest terms.. x 9 x x x x x 4 x ab b a 8ab 9. a 3 a a (x y) (a b) 0. a (x y )(a ab b ). x 3 y 3 (x y) 3. x 4 y 4 (x 4 y x y 3 x 3 y xy 4 )(x y) 5 x x 5 x x 0 x 7x 4 a b ax bx a 3 b 7b 5 (ab 3b ) In Exercises 3 55, carry out the indicated operations and simplify where possible. 3. # x 4 ax 3 4. a a x 3ax x x 4a x x 5. # x 3x x x x 4x 4 6. (3t 4tx x ) 3t tx x t x x 3 y 3 7. x 4xy 3y (x y) 3 x xy 3y a a 4 8. a 4 6a a 49 a 3 6a 36a x 5x x x 30x a b a a 6 c x 4 4 x 3 3 x x 3x x x 6 x 4 x x x a x ax 3ax (x ) (x ) 3 a 5a 4 a 6 a b a b a a b 3 x 5 x x x x 0 x 8x 6 4 6x 5x 4 3x 4x x q p p 9pq 5q a a 8a b b a p q p 5pq x x a x x x 3 4 x a a 38. x a a a b a 4. a b a 4. 3 x h 3 h x h h a a x x a x a x y x y a x x a a ax x

28 B-8 Appendix B x xy y x (x ) (x x) 48. y x x y 49. a 50. (a a ) a a b 5. x(x y) y(x y) 5. x(x y) y(y x) B a b a b a b a b x a 53. # ab3 a 3 b 54. a b a b a b x a a b a b x a x a C 55. a a b b a a a b b b a b a b a a b a b b 56. Consider the three fractions b c bc, c a ca, and a b ab (a) If a, b, and c 3, find the sum of the three fractions. Also compute their product. What do you observe? Show that the sum and the product of the three given fractions are, in fact, always equal. Today s familiar plus and minus signs were first used in 5th-century Germany as warehouse marks. They indicated when a container held something that weighed over or under a certain standard weight. Martin Gardner in Mathematical Games (Scientific American, June 977) B.6 PROPERTIES OF THE REAL NUMBERS I do not like as a symbol for multiplication, as it is easily confounded with x;... often I simply relate two quantities by an interposed dot.... G. W. Leibniz in a letter dated July 9, 698 In this section we will first list the basic properties for the real number system. Then we will use those properties to prove the familiar rules of algebra for working with signed numbers. The set of real numbers is closed with respect to the operations of addition and multiplication. This means that when we add or multiply two real numbers, the result (that is, the sum or the product) is again a real number. Some of the other most basic properties and definitions for the real-number system are listed in the following box. In the box, the lowercase letters a, b, and c denote arbitrary real numbers. PROPERTY SUMMARY Some Fundamental Properties of the Real Numbers Commutative properties a b b a ab ba Associative properties a (b c) (a b) c a(bc) (ab)c Identity properties. There is a unique real number 0 (called zero or the additive identity) such that Inverse properties a 0 a and 0 a a. There is a unique real number (called one or the multiplicative identity) such that a # a and # a a. For each real number a there is a real number a (called the additive inverse of a or the opposite of a) such that a (a) 0 and (a) a 0. For each real number a 0 there is a real number denoted by (or a or a ) a and called the multiplicative inverse or reciprocal of a, such that a # a and a # a Distributive properties a(b c) ab ac (b c)a ba ca

29 B.6 Properties of the Real Numbers B-9 On reading this list of properties for the first time, many students ask the natural question, Why do we even bother to list such obvious properties? One reason is that all the other laws of arithmetic and algebra (including the not-so-obvious ones) can be derived from our rather short list. For example, the rule 0 # a 0 can be proved by using the distributive property, as can the rule that the product of two negative numbers is a positive number. We now state and prove those properties and several others. Theorem Let a and b be real numbers. Then (a) a # 0 0 (c) (a) a (e) (a) ab a ()a (d) a ab Proof of Part (a) Now since a 0 is a real number, it has an additive inverse, (a 0). Adding this to both sides of the last equation, we obtain 0 a # 0 additive identity property Thus a # 0 0, as we wished to show. Proof of Part # a # 0 a # (0 0) additive identity property a # 0 a # 0 a # 0 a # 0 [(a # 0)] (a # 0 a # 0) [(a # 0)] 0 0 # a [ ()]a # a ()a a ()a Now, by adding a to both sides of this last equation, we obtain a 0 a [a ()a] a (a a) ()a additive identity property and a 0 ()a a ()a distributive property a # 0 [(a # 0)] a # 0 5a # 0 [(a # 0)]6 0 a # 0 0 additive inverse property using part (a) and the commutative of multiplication additive inverse property distributive property associative property of addition multiplicative identity property associative property of addition additive inverse property additive identity property This last equation asserts that a ()a, as we wished to show. #

30 B-30 Appendix B Proof of Part (c) (a) (a) 0 additive inverse property By adding a to both sides of this last equation, we obtain [(a) (a)] a 0 a (a) (a a) a associative property of addition and additive identity property (a) 0 a additive inverse property (a) a additive identity property This last equation states that (a) a, as we wished to show. Proof of Part (d) a a[()b] using part [a()]b associative property of multiplication [()a]b commutative property of multiplication ()(ab) associative property of multiplication (ab) using part Thus a ab, as we wished to show. Proof of Part (e) (a) [(a)b] using part (d) [b(a)] commutative property of multiplication [(ba)] using part (d) ba using part (c) ab commutative property of multiplication We ve now shown that (a) ab, as required.