A.1 Radicals and Rational Exponents

Transcription

1 APPENDIX A. Radicals and Rational Eponents 779 Appendies Overview This section contains a review of some basic algebraic skills. (You should read Section P. before reading this appendi.) Radical and rational epressions are introduced and radical epressions are simplified algebraically. We add, subtract, and multiply polynomials and factor simple polynomials by a variety of techniques. These factoring techniques are used to add, subtract, multiply, and divide fractional epressions. What you ll learn about Radicals Simplifying Radical Epressions Rationalizing the Denominator Rational Eponents... and why You need to review these basic algebraic skills if you don t remember them. A. Radicals and Rational Eponents Radicals If b a, then b is a square root of a. For eample, both and - are square roots of 4 because - 4. Similarly, b is a cube root of a if b 3 a. For eample, is a cube root of 8 because 3 8. DEFINITION Real nth Root of a Real Number Let n be an integer greater than and a and b real numbers.. If b n a, then b is an nth root of a.. If a has an nth root, the principal nth root of a is the nth root having the same sign as a. The principal nth root of a is denoted by the radical epression n a. The positive integer n is the inde of the radical and a is the radicand. Every real number has eactly one real nth root whenever n is odd. For instance, is the only real cube root of 8. When n is even, positive real numbers have two real nth roots and negative real numbers have no real nth roots. For eample, the real fourth roots of 6 are, and -6 has no real fourth roots. The principal fourth root of 6 is. When n, special notation is used for roots. We omit the inde and writea instead of a. If a is a positive real number and n a positive even integer, its two nth roots are denoted by n a and - n a. Principal nth Roots and Calculators Most calculators have a key for the principal nth root. Use this feature of your calculator to check the computations in Eample. EXAMPLE Finding Principal nth Roots (a) 36 6 because (b) because a 3 3 A b 7 8. (c) 3 3 because a - b 7 - A (d) 4-65 is not a real number because the inde 4 is even and the radicand -65 is negative (there is no real number whose fourth power is negative). Now try Eercises 7 and 9. Here are some properties of radicals together with eamples that help illustrate their meaning.

2 780 APPENDIX A Caution Without the restriction that preceded the list, property 5 would need special attention. For eample, because -3 Z -3-3 on the right is not a real number. Properties of Radicals Let u and v be real numbers, variables, or algebraic epressions, and m and n be positive integers greater than. We assume that all of the roots are real numbers and all of the denominators are not zero. Property Eample. n uv n u # n v 75 5 # 3 5 # n u Av n u n v 3 m n u m # n u n u n u n u m n u m 6. n u n e ƒ u ƒ n even u n odd A # ƒ -6ƒ Simplifying Radical Epressions Many simplifying techniques for roots of real numbers have been rendered obsolete because of calculators. For eample, when determining the decimal form of /, it was once very common first to change the fraction so that the radical was in the numerator: # Using paper and pencil, it was then easier to divide a decimal approimation for by than to divide that decimal into. Now either form is quickly computed with a calculator. However, these techniques are still valid for radicals involving algebraic epressions and for numerical computations when you need eact answers. Eample illustrates the technique of removing factors from radicands. Properties of Eponents Check the Properties of Eponents on page 7 of Section P. to see why 6 4 and EXAMPLE Removing Factors from Radicands (a) # 5 Finding greatest fourth-power factor 4 4 # # 4 5 Property 4 5 Property 6 (b) # Finding greatest square factor 3 # Properties and 6 (c) 4 4 y 4 4 y 4 ƒyƒ Finding greatest fourth-power factor Property 6 (d) 3-4y 6 3 -y 3 # 3 -y 3 3 Finding greatest cube factor Properties and 6 Now try Eercises 9 and 33.

3 APPENDIX A. Radicals and Rational Eponents 78 Rationalizing the Denominator The process of rewriting fractions containing radicals so that the denominator is free of radicals is rationalizing the denominator. When the denominator has the form n u k, multiplying numerator and denominator by n u n-k and using property 6 will eliminate the radical from the denominator because n u k # n u n-k n u k+n-k n u n. Eample 3 illustrates the process. EXAMPLE 3 Rationalizing the Denominator (a) (b) (c) A 3 3 # # ƒƒ 5 B y y 5 # 5 y 3 5 y 3 5 y y 5 y Rational Eponents 3 Now try Eercise 37. We know how to handle eponential epressions with integer eponents (see Section P.). For eample, 3 # 4 7, 3 6, 5 / 3, - /, and so forth. But eponents can also be rational numbers. How should we define /? If we assume that the same rules that apply for integer eponents also apply for rational eponents we get a clue. For eample, we want / # /. This equation suggests that /. In general, we have the following definition. y y DEFINITION Rational Eponents Let u be a real number, variable, or algebraic epression, and n an integer greater than. Then u /n n u. If m is a positive integer, m/n is in reduced form, and all roots are real numbers, then u m/n u /n m n u m and u m/n u m /n n u m. The numerator of a rational eponent is the power to which the base is raised, and the denominator is the root to be taken. The fraction m/n needs to be in reduced form because, for instance, u /3 3 u is defined for all real numbers u (every real number has a cube root), but u 4/6 6 u 4 is defined only for u Ú 0 (only nonnegative real numbers have sith roots).

4 78 APPENDIX A Simplifying Radicals If you also want the radical form in Eample 4d to be simplified, then continue as follows: z # z 3 z 3 z z z EXAMPLE 4 Converting Radicals to Eponentials and Vice Versa (a) (c) + y 3 + y 3/ /3 y /3 y /3 3 y (b) (d) 35 3 # /5 3 7/5 z -3/ z 3/ z 3 Now try Eercises 43 and 47. An epression involving powers is simplified if each factor appears only once and all eponents are positive. Eample 5 illustrates. EXAMPLE 5 Simplifying Eponential Epressions (a) y 9 /3 y /3 y 3 y 5/3 y 5 /3 -/ /6 3 6 (b) a ba b / /5 y y y 9/0 Now try Eercise 6. Eample 6 suggests how to simplify a sum or difference of radicals. EXAMPLE 6 Combining Radicals (a) (b) # 5-5 # y - y 3 y - y y ƒƒ y - ƒyƒ y ƒƒ - ƒyƒy Find greatest square factors. Remove factors from radicands. Distributive property Find greatest square factors. Remove factors from radicands. Distributive property. Now try Eercise 7. Here s a summary of the procedures we use to simplify epressions involving radicals. Simplifying Radical Epressions. Remove factors from the radicand (see Eample ).. Eliminate radicals from denominators and denominators from radicands (see Eample 3). 3. Combine sums and differences of radicals, if possible (see Eample 6). APPENDIX A. EXERCISES In Eercises 6, find the indicated real roots.. Square roots of 8. Fourth roots of 8 3. Cube roots of Fifth roots of Square roots of 6/9 6. Cube roots of -7/8 In Eercises 7, evaluate the epression without using a calculator A A 5 In Eercises 3, use a calculator to evaluate the epression / 8. 5/ / /3 -/3. a -. a - 8 b b -/3

5 APPENDIX A. Radicals and Rational Eponents 783 In Eercises 3 6, use the information from the grapher screens below to evaluate the epression..5^3 4.4^ In Eercises 7 36, simplify by removing factors from the radicand y y y y y 9 In Eercises 37 4, rationalize the denominator y B 5 a 3 B 3 y b In Eercises 43 46, convert to eponential form a + b y y 46. y4 y 3 In Eercises 47 50, convert to radical form. 47. a 3/4 b /4 48. y / /3 50. y -3/4 In Eercises 5 56, write using a single radical y ab a 3 a 56. a3 a In Eercises 57 64, simplify the eponential epression. 57. a 3/5 a /3 58. y 4 / a 3/ ^.^ a / 6 a 5/3 b 3/4 3a /3 b 5/4 y /3 b p q 4 / a -8 6 /3 6. y -3 b 7q 3 p 6 / y 6 -/3 / -/3 64. a ba3 /3 6 y -/ y y / b In Eercises 65 74, simplify the radical epression y y 8 z B y B y B 3 4 y B 3 y ab 6 # 5 7a b y y + y 3 In Eercises 75 8, replace ~ with 6,, or 7 to make a true statement ~ ~ / ~ /3 ~ ~ ~- 8. /3 ~ 3 3/ /3 ~ 3-3/4 83. The time t (in seconds) that it takes for a pendulum to complete one cycle is approimately t.l, where L is the length (in feet) of the pendulum. How long is the period of a pendulum of length 0 ft? 84. The time t (in seconds) that it takes for a rock to fall a distance d (in meters) is approimately t 0.45d. How long does it take for the rock to fall a distance of 00 m? 85. Writing to Learn Eplain why n a and a real nth root of a need not have the same value.

6 784 APPENDIX A A. Polynomials and Factoring What you ll learn about Adding, Subtracting, and Multiplying Polynomials Special Products Factoring Polynomials Using Special Products Factoring Trinomials Factoring by Grouping... and why You need to review these basic algebraic skills if you don t remember them. Adding, Subtracting, and Multiplying Polynomials A polynomial in is any epression that can be written in the form a n n + a n- n- + Á + a + a 0, where n is a nonnegative integer, and a n Z 0. The numbers a n-, Á, a, a 0 are real numbers called coefficients. The degree of the polynomial is n and the leading coefficient is a n. Polynomials with one, two, or three terms are monomials, binomials, or trinomials, respectively. A polynomial written with powers of in descending order is in standard form. To add or subtract polynomials, we add or subtract like terms using the distributive property. Terms of polynomials that have the same variable each raised to the same power are like terms. EXAMPLE Adding and Subtracting Polynomials (a) (b) SOLUTION (a) We group like terms and then combine them as follows: (b) We group like terms and then combine them as follows: Now try Eercises 9 and. To epand the product of two polynomials we use the distributive property. Here is what the procedure looks like when we multiply the binomials 3 + and Distributive property Distributive property Product of Product of Product of Product of First terms Outer terms Inner terms Last terms In the above FOIL method for products of binomials, the outer (O) and inner (I ) terms are like terms and can be added to give Multiplying two polynomials requires multiplying each term of one polynomial by every term of the other polynomial. A convenient way to compute a product is to arrange the polynomials in standard form one on top of another so their terms align vertically, as illustrated in Eample.

7 APPENDIX A. Polynomials and Factoring 785 EXAMPLE Multiplying Polynomials in Vertical Form Write in standard form. SOLUTION Add. Thus, Now try Eercise 33. Special Products Certain products provide patterns that will be useful when we factor polynomials. Here is a list of some special products for binomials. Special Binomial Products Let u and v be real numbers, variables, or algebraic epressions.. Product of a sum and a difference: u + vu - v u - v. Square of a sum: u + v u + uv + v 3. Square of a difference: u - v u - uv + v 4. Cube of a sum: u + v 3 u 3 + 3u v + 3uv + v 3 5. Cube of a difference: u - v 3 u 3-3u v + 3uv - v 3 EXAMPLE 3 Using Special Products Epand the products. (a) (b) 5y - 4 5y - 5y y - 40y + 6 (c) - 3y y + 33y - 3y y + 54y - 7y 3 Now try Eercises 3, 5, and 7. Factoring Polynomials Using Special Products When we write a polynomial as a product of two or more polynomial factors we are factoring a polynomial. Unless specified otherwise, we factor polynomials into factors of lesser degree and with integer coefficients in this appendi. A polynomial that cannot be factored using integer coefficients is a prime polynomial.

8 786 APPENDIX A A polynomial is completely factored if it is written as a product of its prime factors. For eample, and are completely factored (it can be shown that + is prime). However, is not completely factored because - 9 is not prime. In fact, and is completely factored. The first step in factoring a polynomial is to remove common factors from its terms using the distributive property as illustrated by Eample 4. EXAMPLE 4 Removing Common Factors (a) is the common factor. (b) u 3 v + uv 3 uvu + v uv is the common factor. Now try Eercise 43. Recognizing the epanded form of the five special binomial products will help us factor them. The special form that is easiest to identify is the difference of two squares. The two binomial factors have opposite signs: Two squares Square roots u - v u + vu - v. Difference Opposite signs EXAMPLE 5 Factoring the Difference of Two Squares (a) (b) 4 - y y y y y y - 3 Difference of two squares Factors are prime. Difference of two squares Factors are prime. Simplify. Now try Eercise 45. A perfect square trinomial is the square of a binomial and has one of the two forms shown here. The first and last terms are squares of u and v, and the middle term is twice the product of u and v. The operation signs before the middle term and in the binomial factor are the same. Perfect square (sum) u + uv + v u + v Perfect square (difference) u - uv + v u - v Same signs Same signs

9 APPENDIX A. Polynomials and Factoring 787 EXAMPLE 6 Factoring Perfect Square Trinomials (a) u 3, v 3 + (b) 4 - y + 9y - 3y + 3y u, v 3y - 3y Now try Eercise 49. In the sum and difference of two cubes, notice the pattern of the signs. Same signs u 3 + v 3 u + vu - uv + v Same signs u 3 - v 3 u - vu + uv + v Opposite signs Opposite signs EXAMPLE 7 Factoring the Sum and Difference of Two Cubes (a) (b) Factoring Trinomials Difference of two cubes Factors are prime. Sum of two cubes Factors are prime. Now try Eercise 55. Factoring the trinomial a + b + c into a product of binomials with integer coefficients requires factoring the integers a and c. Factors of a a + b + c n +nn +n Factors of c Because the number of integer factors of a and c is finite, we can list all possible binomial factors. Then we begin checking each possibility until we find a pair that works. (If no pair works, then the trinomial is prime.) Eample 8 illustrates. EXAMPLE 8 Factoring a Trinomial with Leading Coefficient Factor SOLUTION The only factor pair of the leading coefficient is and. The factor pairs of 4 are and 4, and and 7. Here are the four possible factorizations of the trinomial: If you check the middle term from each factorization you will find that Now try Eercise 59. With practice you will find that it usually is not necessary to list all possible binomial factors. Often you can test the possibilities mentally.

10 788 APPENDIX A EXAMPLE 9 Factoring a Trinomial with Leading Coefficient Factor SOLUTION The factor pairs of the leading coefficient are and 35, and 5 and 7. The factor pairs of are and, and 6, and 3 and 4. The possible factorizations must be of the form - *35 +?, + *35 -?, 5 - *7 +?, 5 + *7 -?, where * and? are one of the factor pairs of. Because the two binomial factors have opposite signs, there are 6 possibilities for each of the four forms a total of 4 possibilities in all. If you try them, mentally and systematically, you should find that ). Now try Eercise 63. We can etend the technique of Eamples 8 and 9 to trinomials in two variables as illustrated in Eample 0. EXAMPLE 0 Factoring Trinomials in and y Factor 3-7y + y. SOLUTION The only way to get -7y as the middle term is with 3-7y + y 3 -?y -?y. The signs in the binomials must be negative because the coefficient of is positive and the coefficient of the middle term is negative. Checking the two possibilities, 3 - y - y and 3 - y - y, shows that 3-7y + y 3 - y - y. Now try Eercise 67. y Factoring by Grouping Notice that a + bc + d ac + ad + bc + bd. If a polynomial with four terms is the product of two binomials, we can group terms to factor. There are only three ways to group the terms and two of them work. So, if two of the possibilities fail, then it is not factorable. EXAMPLE Factoring by Grouping (a) (b) ac - ad + bc - bd ac - ad + bc - bd ac - d + bc - d c - da + b Here is a checklist for factoring polynomials. Group terms. Factor each group. Distributive property Group terms. Factor each group. Distributive property Now try Eercise 69.

11 APPENDIX A. Polynomials and Factoring 789 Factoring Polynomials. Look for common factors.. Look for special polynomial forms. 3. Use factor pairs. 4. If there are four terms, try grouping. APPENDIX A. EXERCISES In Eercises 4, write the polynomial in standard form and state its degree In Eercises 5 8, state whether the epression is a polynomial In Eercises 9 8, simplify the epression. Write your answer in standard form y + y y + 3y y y + 3y u4u v - 3v In Eercises 9 40, epand the product. Use vertical alignment in Eercises 33 and y3 + y y u - v 3 8. u + 3v y 3 + 3y / - y / / + y / 37. u + vu - v In Eercises 4 44, factor out the common factor yz 3-3yz + yz In Eercises 45 48, factor the difference of two squares. 45. z y y In Eercises 49 5, factor the perfect square trinomial. 49. y + 8y y + y z - 4z z - 4z + 6 In Eercises 53 58, factor the sum or difference of two cubes. 53. y z y z y 3 In Eercises 59 68, factor the trinomial y - y z - 5z t + 5t u - 33u v + 3v y + y y - 0y y - 4y In Eercises 69 74, factor by grouping ac + 6ad - bc - 3bd 74. 3uw + uz - vw - 8vz

12 790 APPENDIX A In Eercises 75 90, factor completely y 3-0y + 5y 77. 8y y + 3y y - y y + 3y - y 3 8. z - 8z y - 0y 87. ac - bd + 4ad - bc 88. 6ac - bd + 4bc - 3ad Writing to Learn Show that the grouping ac + bc - ad + bd leads to the same factorization as in Eample b. Eplain why the third possibility, ac - bd + -ad + bc does not lead to a factorization.

13 APPENDIX A.3 Fractional Epressions 79 A.3 Fractional Epressions What you ll learn about Domain of an Algebraic Epression Reducing Rational Epressions Operations with Rational Epressions Compound Rational Epressions... and why You need to review these basic algebraic skills if you don t remember them. Domain of an Algebraic Epression A quotient of two algebraic epressions, besides being another algebraic epression, is a fractional epression, or simply a fraction. If the quotient can be written as the ratio of two polynomials, the fractional epression is a rational epression. Here are eamples of each The one on the left is a fractional epression but not a rational epression. The other is both a fractional epression and a rational epression. Unlike polynomials, which are defined for all real numbers, some algebraic epressions are not defined for some real numbers. The set of real numbers for which an algebraic epression is defined is the domain of the algebraic epression. EXAMPLE Finding Domains of Algebraic Epressions (a) (b) - (c) - SOLUTION (a) The domain of , like that of any polynomial, is the set of all real numbers. (b) Because only nonnegative numbers have square roots, - Ú 0, or Ú. In interval notation, the domain is 3, q. (c) Because division by zero is undefined, - Z 0, or Z. The domain is the set of all real numbers ecept. Now try Eercises and 3. Reducing Rational Epressions Let u, v, and z be real numbers, variables, or algebraic epressions. We can write rational epressions in simpler form using uz vz u v provided z Z 0. This requires that we first factor the numerator and denominator into prime factors. When all factors common to numerator and denominator have been removed, the rational epression (or rational number) is in reduced form. EXAMPLE Reducing Rational Epressions Write - 3/ - 9 in reduced form. SOLUTION , Z 3 Factor completely. Remove common factors. We include Z 3 as part of the reduced form because 3 is not in the domain of the original rational epression and thus should not be in the domain of the final rational epression. Now try Eercise 35.

14 79 APPENDIX A Two rational epressions are equivalent if they have the same domain and have the same value for all numbers in the domain. The reduced form of a rational epression must have the same domain as the original rational epression. This is why we attached the restriction Z 3 to the reduced form in Eample. Operations with Rational Epressions Two fractions are equal, u/v z /w, if and only if uw vz. Here is how we operate with fractions. Invert and Multiply The division step shown in 4 is often referred to as invert the divisor (the fraction following the division symbol) and multiply the result times the numerator (the first fraction). Operations with Fractions Let u, v, w, and z be real numbers, variables, or algebraic epressions. All of the denominators are assumed to be different from zero. Operation Eample u v + w v u + w v 3 3 u # # 4 3 # v + w uz + vw z vz 5 5 u # w v z uw vz u v, w z u # z v w uz vw 5. For subtraction, replace + by - in and. # # 4 3 # , 4 5 # EXAMPLE 3 Multiplying and Dividing Rational Epressions + - (a) # # Factor completely , Z, Z -7 Remove common factors. 3 + (b) - -, Invert and multiply Factor completely , Z -, Z Remove common factors. Now try Eercises 49 and 55. Note on Eample The numerator, + 4-6, of the final epression in Eample 4 is a prime polynomial. Thus, there are no common factors. EXAMPLE 4 Adding Rational Epressions Definition of addition Distributive property Combine like terms. Now try Eercise 59.

15 APPENDIX A.3 Fractional Epressions 793 If the denominators of fractions have common factors, then it is often more efficient to find the LCD before adding or subtracting the fractions. The LCD (least common denominator) is the product of all the prime factors in the denominators, where each factor is raised to the greatest power found in any one denominator for that factor. EXAMPLE 5 Using the LCD Write the following epression as a fraction in reduced form SOLUTION The factored denominators are -,, and - +, respectively. The LCD is - + ) , Z Factor. Equivalent fractions Combine numerators. Epand terms. Simplify. Factor. Reduce. Now try Eercise 6. Compound Rational Epressions Sometimes a complicated algebraic epression needs to be changed to a more familiar form before we can work on it. A compound fraction (sometimes called a comple fraction), in which the numerators and denominators may themselves contain fractions, is such an eample. One way to simplify a compound fraction is to write both the numerator and denominator as single fractions and then invert and multiply. If the fraction then takes the form of a rational epression, we write the epression in reduced or simplest form. EXAMPLE 6 Simplifying a Compound Fraction Combine fractions Simplify , Z 3 Invert and multiply. Now try Eercise 63.

16 794 APPENDIX A A second way to simplify a compound fraction is to multiply the numerator and denominator by the LCD of all fractions in the numerator and denominator as illustrated in Eample 7. EXAMPLE 7 Simplifying Another Compound Fraction Use the LCD to simplify the compound fraction SOLUTION The LCD of the four fractions in the numerator and denominator is a b. a - b a a - b ba b Multiply numerator and a - a b a - b ba b denominator by LCD. b - a ab - a b b + ab - a abb - a b + a ab, a Z b a - b a -. b Simplify. Factor. Reduce. Now try Eercise 69. APPENDIX A.3 EXERCISES In Eercises 8, rewrite as a single fraction # # , 5 3, In Eercises 9 8, find the domain of the algebraic epression , Z , Z In Eercises 9 6, find the missing numerator or denominator so that the two rational epressions are equal y 5y 3? 3? ? - 4? ? ? ? - 3 +? In Eercises 7 3, consider the original fraction and its reduced form from the specified eample. Eplain why the given restriction is needed on the reduced form. 7. Eample 3a, Z, Z Eample 3b, Z -, Z 9. Eample 4, none 30. Eample 5, Z 0 3. Eample 6, Z 3 3. Eample 7, a Z b

17 APPENDIX A.3 Fractional Epressions 795 In Eercises 33 44, write the epression in reduced form. y - y y y 5 9y y y - 3 y + 6y y - 4y + + z - 3z z y y y 3-3y y 3 + 4y y y z 3 - z 3 + 6z + 8z In Eercises 63 70, simplify the compound fraction. z + 5z - 3 z y + 3y + y - y y y 3 + 3y - 5y - 5 In Eercises 45 6, simplify. - y - y # # y 47. # # + h - + h h y y 3 + y + 4y # y h h - 4 # 4 b + + y 3 + y y 3-8 a + a - a b b y + 9y - 5 y - 5 y + 8y # 3y + y b # y - 5 y - y 3y - y - y + 4 a - a a - b b y, 8y, In Eercises 7 74, write with positive eponents and simplify. 4 + y y 4-4y 55., y 7. a 7. + b + y- y 56., - y - 4y 3y 4y 3y y y - -