7.2 Rational Exponents
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1 Section 7.2 Rational Exponents Rational Exponents S Understand the Meaning of a /n. 2 Understand the Meaning of a m/n. 3 Understand the Meaning of a -m/n. 4 Use Rules for Exponents to Simplify Expressions That Contain Rational Exponents. 5 Use Rational Exponents to Simplify Radical Expressions. Understanding the Meaning of a /n So far in this text, we have not defined expressions with rational exponents such as 3 /2, x 2/3, and -9 -/4. We will define these expressions so that the rules for exponents will apply to these rational exponents as well. Suppose that x = 5 /3. Then x 3 = 5 /3 2 3 = 5 /3 # 3 = 5 or 5 using rules for exponents Since x 3 = 5, x is the number whose cube is 5, or x = Notice that we also know that x = 5 /3. This means 5 /3 = Definition of a /n T If n is a positive integer greater than and 2 n a is a real number, then a /n = 2 n a Notice that the denominator of the rational exponent corresponds to the index of the radical. EXAMPLE Use radical notation to write the following. Simplify if possible. a. 4 /2 b. 64 /3 c. x /4 d. 0 /6 e. -9 /2 f. 8x 8 2 /4 g. 5y /3 a. 4 /2 = 24 = 2 b. 64 /3 = = 4 c. x /4 = 2 4 x d. 0 /6 = = 0 e. -9 /2 = - 29 = -3 f. (8x 8 ) /4 = 2 4 8x 8 = 3x 2 g. 5y /3 = 52 3 y Use radical notation to write the following. Simplify if possible. a. 36 /2 b. 000 /3 c. x /3 d. /4 e. -64 /2 f. 25x 9 2 /3 g. 3x /4 2 Understanding the Meaning of a m/n As we expand our use of exponents to include m, we define their meaning so that n rules for exponents still hold true. For example, by properties of exponents, 8 2/3 = 8 /3 2 2 = or 8 2/3 = /3 = Definition of a m/n If m and n are positive integers greater than with m n in simplest form, then as long as 2 n a is a real number. a m/n = 2 n a m = 2 n a2 m
2 420 CHAPTER 7 Rational Exponents, Radicals, and Complex Numbers Notice that the denominator n of the rational exponent corresponds to the index of the radical. The numerator m of the rational exponent indicates that the base is to be raised to the mth power. This means 8 2/3 = = = 4 or 8 2/3 = = 2 2 = 4 From simplifying 8 2/3, can you see that it doesn t matter whether you raise to a power first and then take the nth root or you take the nth root first and then raise to a power? Helpful Hint Most of the time, 2 n a2 m will be easier to calculate than 2 n a m. EXAMPLE 2 Use radical notation to write the following. Then simplify if possible. a. 4 3/2 b. -6 3/4 c /3 d. a 9 b 3/2 e. 4x - 2 3/5 a. 4 3/2 = = 2 3 = 8 b. -6 3/4 = = = -8 c /3 = = = 9 d. a 9 b 3/2 e. 4x - 2 3/5 = 2 5 4x = a A 9 b 3 = a 3 b 3 2 Use radical notation to write the following. Simplify if possible. a. 6 3/2 b. - 3/5 c /4 d. a 3/2 25 b e. 3x /9 = 27 Helpful Hint The denominator of a rational exponent is the index of the corresponding radical. For example, x /5 = 2 5 x and z 2/3 = 2 3 z 2, or z 2/3 = 2 3 z Understanding the Meaning of a m/n The rational exponents we have given meaning to exclude negative rational numbers. To complete the set of definitions, we define a -m/n. Definition of a m/n a -m/n = as long as a m/n is a nonzero real number. a m/n EXAMPLE 3 Write each expression with a positive exponent, and then simplify. a. 6-3/4 b /3 a. 6-3/4 = 6 3/4 = = 2 3 = 8 b /3 = /3 = = = 9
3 Section 7.2 Rational Exponents 42 3 Write each expression with a positive exponent; then simplify. a. 9-3/2 b /3 Helpful Hint If an expression contains a negative rational exponent, such as 9-3/2, you may want to first write the expression with a positive exponent and then interpret the rational exponent. Notice that the sign of the base is not affected by the sign of its exponent. For example, 9-3/2 = 9 = 3/2 292 = 3 27 Also, /3 = -272 = - /3 3 CONCEPT CHECK Which one is correct? a. -8 2/3 = 4 b. 8-2/3 = - 4 c. 8-2/3 = -4 d. -8-2/3 = Using Rules for Exponents to Simplify Expressions It can be shown that the properties of integer exponents hold for rational exponents. By using these properties and definitions, we can now simplify expressions that contain rational exponents. These rules are repeated here for review. Note: For the remainder of this chapter, we will assume that variables represent positive real numbers. Since this is so, we need not insert absolute value bars when we simplify even roots. Summary of Exponent Rules If m and n are rational numbers, and a, b, and c are numbers for which the expressions below exist, then Product rule for exponents: a m # a n = a m +n Power rule for exponents: a m 2 n = a m # n Power rules for products and quotients: ab2 n = a n b n and a a n c b = an c n, c 0 Quotient rule for exponents: am a n = am -n, a 0 Zero exponent: a 0 =, a 0 Negative exponent: a -n = a n, a 0 Answer to Concept Check: d EXAMPLE 4 Use properties of exponents to simplify. Write results with only positive exponents. a. b /3 # b 5/3 b. x /2 x /3 c. 7/3 d. y -4/7 2x # 2/5 y -/3 2 5 y 6/7 e. x 2 y (Continued on next page) 7 4/3
4 422 CHAPTER 7 Rational Exponents, Radicals, and Complex Numbers a. b /3 # b 5/3 = b /3 +5/32 = b 6/3 = b 2 b. x /2 x /3 = x /2 +/32 = x 3/6 +2/6 = x 5/6 Use the product rule. c. 7/3 7 4/3 = 7/3-4/3 = 7-3/3 = 7 - = Use the quotient rule. 7 d. y - 4/7 # y 6/7 = y - 4/7 +6/7 = y 2/7 Use the product rule. e. We begin by using the power rule ab2 m = a m b m to simplify the numerator. 2x 2/5 y -/3 2 5 x 2 y = 25 x 2/5 2 5 y -/3 2 5 x 2 y = 32x /3-3/3 y = 32x2 y -5/3 x 2 y Use the power rule and simplify Apply the quotient rule. = 32x 0 y -8/3 = 32 y 8/3 4 Use properties of exponents to simplify. a. y 2/3 # y 8/3 b. x 3/5 # x /4 c. 92/7 d. b 4/9 # b -2/9 e. 3x/4 y -2/3 2 4 x 4 y 9 9/7 EXAMPLE 5 Multiply. a. z 2/3 z /3 - z 5 2 b. x /3-52x / a. z 2/3 z /3 - z 5 2 = z 2/3 z /3 - z 2/3 z 5 = z 2/3 +/32 2/ z = z 3/3 2/3 +5/32 - z = z - z 7/3 Apply the distributive property. Use the product rule. b. x /3-52x / = x 2/3 + 2x /3-5x /3-0 = x 2/3-3x /3-0 5 Multiply. a. x 3/5 x /3 - x 2 2 b. x /2 + 62x /2-22 Think of x /3-52 and x / as 2 binomials, and FOIL. EXAMPLE 6 Factor x -/2 from the expression 3x -/2-7x 5/2. Assume that all variables represent positive numbers. 3x -/2-7x 5/2 = x -/ x -/2 27x 6/2 2 = x -/2 3-7x 3 2 To check, multiply x -/2 3-7x 3 2 to see that the product is 3x -/2-7x 5/2. 6 Factor x -/5 from the expression 2x -/5-7x 4/5.
5 Section 7.2 Rational Exponents Using Rational Exponents to Simplify Radical Expressions Some radical expressions are easier to simplify when we first write them with rational exponents. We can simplify some radical expressions by first writing the expression with rational exponents. Use properties of exponents to simplify, and then convert back to radical notation. EXAMPLE 7 Use rational exponents to simplify. Assume that variables represent positive numbers. a. 2 8 x 4 b c. 2 4 r 2 s 6 a. 2 8 x 4 = x 4/8 = x /2 = 2x b = 25 /6 = /6 = 5 2/6 = 5 /3 = c. 2 4 r 2 s 6 = r 2 s 6 2 /4 = r 2/4 s 6/4 = r /2 s 3/2 = rs 3 2 /2 = 2rs 3 7 Use rational exponents to simplify. Assume that the variables represent positive numbers. a. 2 9 x 3 b c. 2 8 a 4 b 2 EXAMPLE 8 Use rational exponents to write as a single radical. a. 2x # 2 4 x b. 2x c # x a. 2x # 2 4 x = x /2 # x /4 /2 +/4 = x = x 3/4 = 2 4 x 3 b. 2x 2 3 x = x/2 x /3 = x/2 -/3 3/6-2/6 = x = x /6 = 2 6 x c # 22 = 3 /3 # 2 /2 = 3 2/6 # 2 3/6 Write with rational exponents. Write the exponents so that they have the same denominator. = 3 2 # /6 = # 2 3 = Use a n b n = ab2 n Write with radical notation. Multiply 3 2 # Use rational expressions to write each of the following as a single radical. a. 2 3 x # 2 4 x b. 23 y c # y Vocabulary, Readiness & Video Check Answer each true or false.. 9 -/2 is a positive number /2 is a whole number. 3. a -m/n = am/n (where a m/n is a nonzero real number).
6 424 CHAPTER 7 Rational Exponents, Radicals, and Complex Numbers Fill in the blank with the correct choice. 4. To simplify x 2/3 # x /5, the exponents. a. add b. subtract c. multiply d. divide 5. To simplify x 2/3 2 /5, the exponents. a. add b. subtract c. multiply d. divide 6. To simplify x2/3 x /5, the exponents. a. add b. subtract c. multiply d. divide Martin-Gay Interactive Videos See Video 7.2 Watch the section lecture video and answer the following questions From looking at Example 2, what is -3x2 /5 in radical notation? 8. From Examples 3 and 4, in a fractional exponent, what do the numerator and denominator each represent in radical form? 9. Based on Example 5, complete the following statements. A negative fractional exponent will move a base from the numerator to the with the fractional exponent becoming. 0. Based on Examples 7 9, complete the following statements. Assume you have an expression with fractional exponents. If applying the product rule of exponents, you the exponents. If applying the quotient rule of exponents, you the exponents. If applying the power rule of exponents, you the exponents.. From Example 0, describe a way to simplify a radical of a variable raised to a power if the index and the exponent have a common factor. 7.2 Exercise Set Use radical notation to write each expression. Simplify if possible. See Example.. 49 / / / /3 5. a 6 b /4 6. a 64 b / / /4 9. 2m /3 0. 2m2 /3. 9x 4 2 /2 2. 6x 8 2 / / / / /5 Use radical notation to write each expression. Simplify if possible. See Example / / / / / / x2 3/ x 3/ x /3 26. x /4 27. a 6 9 b 3/2 28. a b 3/2 Write with positive exponents. Simplify if possible. See Example / / / / / /4 35. x -/4 36. y -/ a -2/3 n -8/ x -3/4 3y -5/7 Use the properties of exponents to simplify each expression. Write with positive exponents. See Example a 2/3 a 5/3 42. b 9/5 b 8/5 43. x -2/5 # x 7/5 44. y 4/3 # y -/ /4 # 3 3/ /2 # 5 /6 y /3 x 3/ y /6 x / u 2 2 3/ /5 x 2/ b /2 b 3/4 -b /4 52. x 3 2 /2 x 7/2 54. a /4 a -/2 a 2/3 y /3 y 5 2 /3
7 Section 7.2 Rational Exponents x /4 2 3 x /2 56. y 3 z2 /6 y -/2 z /3 58. x 3 y 2 2 /4 x -5 y - 2 -/ x /5 2 4 x 3/0 m 2 n2 /4 m -/2 n 5/8 a -2 b 3 2 /8 a -3 b2 -/4 Multiply. See Example y /2 y /2 - y 2/ x /2 x /2 + x 3/ x 2/3 x x /2 x + y x / x / y /2 + 52y / Factor the given factor from the expression. See Example x 8/3 ; x 8/3 + x 0/3 68. x 3/2 ; x 5/2 - x 3/2 69. x /5 ; x 2/5-3x /5 70. x 2/7 ; x 3/7-2x 2/7 7. x -/3 ; 5x -/3 + x 2/3 72. x -3/4 ; x -3/4 + 3x /4 Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. See Example x a 3 Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as 8 # 3 because 8 is a perfect cube CONCEPT EXTENSIONS Choose the correct letter for each exercise. Letters will be used more than once. No pencil is needed. Just think about the meaning of each expression. A = 2, B = -2, C = not a real number / / / /3. -8 / /3 Basal metabolic rate (BMR) is the number of calories per day a person needs to maintain life. A person s basal metabolic rate B(w) in calories per day can be estimated with the function Bw2 = 70w 3/4, where w is the person s weight in kilograms. Use this information to answer Exercises 3 and Estimate the BMR for a person who weighs 60 kilograms. Round to the nearest calorie. (Note: 60 kilograms is approximately 32 pounds.) x y x 4 y y 6 z a 8 b a 5 b x y Use rational expressions to write as a single radical expression. See Example y # 2 5 y y 2 # 2 6 y b b a 2 5 a x # 2 4 x # 2 8 x y # 2 3 y # 2 5 y a a b 2 0 2b # # # 2 3 y # 2 3 x r # 2 3 s b # 2 5 4a REVIEW AND PREVIEW Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 8 as 9 # 2 because 9 is a perfect square Estimate the BMR for a person who weighs 90 kilograms. Round to the nearest calorie. (Note: 90 kilograms is approximately 98 pounds.) The number of cell telephone subscribers in the United States from can be modeled by f x2 = 25x 23/25, where f(x) is the number of cellular telephone subscriptions in millions, x years after 995. (Source: CTIA-Wireless Association, ) Use this information to answer Exercises 5 and Use this model to estimate the number of cellular subscriptions in 200. Round to the nearest tenth of a million. 6. Predict the number of cellular telephone subscriptions in 205. Round to the nearest tenth of a million. 7. Explain how writing x -7 with positive exponents is similar to writing x ->4 with positive exponents. 8. Explain how writing 2x -5 with positive exponents is similar to writing 2x -3>4 with positive exponents. Fill in each box with the correct expression. 9. # a 2/3 = a 3/3, or a 20. # x /8 = x 4/8, or x /2
8 426 CHAPTER 7 Rational Exponents, Radicals, and Complex Numbers 2. = x3/5-2/5 x 22. y = -3/4 y4/4, or y Use a calculator to write a four-decimal-place approximation of each number / / / /7 27. In physics, the speed of a wave traveling over a stretched string with tension t and density u is given by the expression 2t. Write this expression with rational exponents. 2u 28. In electronics, the angular frequency of oscillations in a certain type of circuit is given by the expression LC2 -/2. Use radical notation to write this expression. 7.3 Simplifying Radical Expressions S Use the Product Rule for Radicals. 2 Use the Quotient Rule for Radicals. 3 Simplify Radicals. 4 Use the Distance and Midpoint Formulas. Using the Product Rule It is possible to simplify some radicals that do not evaluate to rational numbers. To do so, we use a product rule and a quotient rule for radicals. To discover the product rule, notice the following pattern. 29 # 24 = 3 # 2 = 6 29 # 4 = 236 = 6 Since both expressions simplify to 6, it is true that 29 # 24 = 29 # 4 This pattern suggests the following product rule for radicals. Product Rule for Radicals If 2 n a and 2 n b are real numbers, then 2 n a # 2 n b = 2 n ab Notice that the product rule is the relationship a /n # b /n = ab2 /n stated in radical notation. EXAMPLE Multiply. a. 23 # 25 b. 22 # 2x c # d y 2 # 2 4 2x 3 2 e. # b A a A 3 a. 23 # 25 = 23 # 5 = 25 b. 22 # 2x = 22x c # = # 2 = = 2 d y 2 # 2 4 2x 3 = 2 4 5y 2 # 2x 3 = 2 4 0y 2 x 3 2 e. # b A a A 3 = 2 # b A a 3 = 2b A 3a Multiply. a. 25 # 27 b. 23 # 2z c # d y # 2 3 3x 2 5 e. # t A m A 2
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