Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have


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1 8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents 4. Write an expression in radical or exponential form In Section 8.1, we discussed the radical notation, along with the concept of roots. In this section, we use that concept to develop a new notation, using exponents that provide an alternate way of writing these roots. That new notation involves rational numbers as exponents. To start the development, we extend all the previous properties of exponents to include rational exponents. Given that extension, suppose that a (1) Squaring both sides of the equation yields a 2 (4 1 2 ) 2 or NOTE We will see later in this chapter that the property (x m ) n x mn holds for rational numbers m and n. a 2 4 (1 2)(2) a a 2 4 (2) From equation (2) we see that a is the number whose square is 4; that is, a is the principal square root of 4. Using our earlier notation, we can write a 14 But from (1) a and to be consistent, we must have NOTE indicates the principal square root of This argument can be repeated for any exponent of the form so it seems reasonable to n, make the following definition. Definitions: If a is any real number and n is a positive integer (n 1), then a 1 n 1 n a Rational Exponents We restrict a so that a is nonnegative when n is even. In words, a 1 n indicates the principal nth root of a. Example 1 illustrates the use of rational exponents to represent roots. 635
2 636 CHAPTER 8 RADICAL EXPRESSIONS Example 1 Writing Expressions in Radical Form NOTE is the cube root of 27. NOTE is the fifth root of 32. Write each expression in radical form and then simplify. (a) (b) (c) (d) ( 36) is not a real number. (e) CHECK YOURSELF 1 Write each expression in radical form and simplify. (a) (b) (c) We are now ready to extend our exponent notation to allow any rational exponent, again assuming that our previous exponent properties must still be valid. Note that NOTE This is because m n (m) 1 n 1 n (m) a m n (a 1 n ) m (a m ) 1 n From our earlier work, we know that a 1 n 1 n a, and combining this with the above observation, we offer the following definition for a m n. Definitions: NOTE The two radical forms for a m n are equivalent, and the choice of which form to use generally depends on whether we are evaluating numerical expressions or rewriting expressions containing variables in radical form. For any real number a and positive integers m and n with n 1, a m n (1 n a) m 2 n a m This new extension of our rational exponent notation is applied in Example 2. Example 2 Simplifying Expressions with Rational Exponents (a) (9 1 2 ) 3 (19) (b) (c) A ( 8) 2 3 (( 8) 1 3 ) 2 (1 3 8) 2 ( 2) 2 4
3 RATIONAL EXPONENTS SECTION NOTE This illustrates why we use (1 n a) m for a m n when evaluating numerical expressions. The numbers involved will be smaller and easier to work with. In (a) we could also have evaluated the expression as CHECK YOURSELF 2 (a) (b) (c) ( 32) Now we want to extend our rational exponent notation. Using the definition of negative exponents, we can write a m n 1 a m n Example 3 illustrates the use of negative rational exponents. Simplifying Expressions with Rational Exponents (a) Example (b) (1 3 27) CHECK YOURSELF 3 (a) (b) Graphing calculators can be used to evaluate expressions that contain rational exponents by using the key and the parentheses keys. Example 4 Estimating Powers Using a Calculator NOTE If you are using a scientific calculator, try using the y x key in place of the key. Using a graphing calculator, evaluate each of the following. Round all answers to three decimal places. (a) Enter 45 and press the ( 2 5) ) key. Then use the following keystrokes: Press ENTER, and the display will read Rounded to three decimal places, the result is
4 638 CHAPTER 8 RADICAL EXPRESSIONS (b) NOTE The ( ) key changes the sign of the exponent to minus. Enter 38 and press the key. Then use the following keystrokes: ( ( ) 2 3 ) Press ENTER, and the display will read Rounded to three decimal places, the result is CHECK YOURSELF 4 Evaluate each of the following by using a calculator. Round each answer to three decimal places. (a) (b) As we mentioned earlier in this section, we assume that all our previous exponent properties will continue to hold for rational exponents. Those properties are restated here. Rules and Properties: Properties of Exponents For any nonzero real numbers a and b and rational numbers m and n, 1. Product rule a m a n a m n a m 2. Quotient rule a n am n 3. Power rule (a m ) n a mn 4. Productpower rule (ab) m a m b m 5. Quotientpower rule a b m am b m We restrict a and b to being nonnegative real numbers when m or n indicates an even root. Example 5 illustrates the use of our extended properties to simplify expressions involving rational exponents. Here, we assume that all variables represent positive real numbers. Example 5 Simplifying Expressions NOTE Product rule add the exponents. NOTE Quotient rule subtract the exponents. NOTE Power rule multiply the exponents. (a) x 2 3 x 1 2 x x x 7 6 w 3 4 (b) 1 2 w w w w 1 4 (c) (a 2 3 ) 3 4 a (2 3)(3 4) a 1 2
5 RATIONAL EXPONENTS SECTION CHECK YOURSELF 5 x 5 6 (a) z 3 4 z 1 2 (b) (c) (b 5 6 ) 2 5 x 1 3 As you would expect from your previous experience with exponents, simplifying expressions often involves using several exponent properties. Example 6 Simplifying Expressions (a) (x 2 3 y 5 6 ) 3 2 (b) (c) (x 2 3 ) 3 2 ( y 5 6 ) 3 2 x (2 3)(3 2) y (5 6)(3 2) xy 5 4 r 1 2 s (r 1 2 ) 6 (s 1 3 ) 6 r 3 s 2 1 r 3 s 2 4a 2 3 b 2 a 1 3 b b 2 b 4 a 1 3 a b 6 a 1 2 (4b6 ) 1 2 a (b 6 ) 1 2 a 1 2 Product power rule. Power rule. Quotientpower rule. Power rule. We simplify inside the parentheses as the first step. 2b3 a 1 2 CHECK YOURSELF 6 w1 2 8x (a) (a 3 4 b 1 2 ) 2 3 (b) (c) 3 4 y x 1 4 y 1 3 z We can also use the relationships between rational exponents and radicals to write expressions involving rational exponents as radicals and vice versa. NOTE Here we use a m n 2 n a m, which is generally the preferred form in this situation. Example 7 Writing Expressions in Radical Form Write each expression in radical form. (a) a a 3 (b) (mn) (mn) m 3 n 3
6 640 CHAPTER 8 RADICAL EXPRESSIONS NOTE Notice that the exponent applies only to the variable y. NOTE Now the exponent applies to 2y because of the parentheses. (c) 2y y 5 (d) (2y) (2y) y 5 CHECK YOURSELF 7 Write each expression in radical form. (a) (ab) 2 3 (b) 3x 3 4 (c) (3x) 3 4 Example 8 Writing Expressions in Exponential Form Using rational exponents, write each expression and simplify. (a) (b) (c) 1 3 5x (5x) a 2 b 4 (9a 2 b 4 ) (a 2 ) 1 2 (b 4 ) 1 2 3ab w 12 z 8 (16w 12 z 8 ) (w 12 ) 1 4 (z 8 ) 1 4 2w 3 z 2 CHECK YOURSELF 8 Using rational exponents, write each expression and simplify. (a) 17a (b) p 6 q 9 (c) x 8 y 16 CHECK YOURSELF ANSWERS 1. (a) 2; (b) 8; (c) (a) 8; (b) ; (c) 8 3. (a) ; (b) (a) 6.562; (b) (a) z 5 4 ; (b) x 1 2 ; (c) b 1 3 2y 2 6. (a) a 1 2 b 1 3 ; (b) w 2 z; (c) 7. (a) 2 3 a 2 b 2 ; (b) 32 4 x 3 ; (c) x 3 x (a) (7a) 1 2 ; (b) 3p 2 q 3 ; (c) 3x 2 y 4
7 Name 8.6 Exercises Section Date In exercises 1 to 12, use the definition of a 1/n to evaluate each expression ( 64) ( 49) ( 64) ANSWERS In exercises 13 to 22, use the definition of a m n to evaluate each expression ( 8) ( 243) In exercises 23 to 32, use the definition of a m n to evaluate the following expressions. Use your calculator to check each answer
8 ANSWERS In exercises 33 to 76, use the properties of exponents to simplify each expression. Assume all variables represent positive real numbers. 33. x 1 2 x a 2 3 a y 3 5 y m 1 4 m b 2 3 b p 5 6 p 2 3 x 2 3 x s 7 5 s w 5 4 w a 5 6 a 1 6 z 9 2 z 3 2 b 7 6 b (x 3 4 ) (y 4 3 ) (a 2 5 ) ( p 3 4 ) (y 3 4 ) (w 2 3 ) (a 2 3 b 3 2 ) (p 3 4 q 5 2 ) (2x 1 5 y 3 5 ) (3m 3 4 n 5 4 ) (s 3 4 t 1 4 ) (x 5 2 y 5 7 ) (8p 3 2 q 5 2 ) (16a 1 3 b 2 3 ) (x 3 5 y 3 4 z 3 2 ) (p 5 6 q 2 3 r 5 3 )
9 ANSWERS a 5 6 b 3 4 x 2 3 y a 1 3 b 1 2 x 1 2 y (r 1 s 1 2 ) r s 1 2 x y m n r 1 2 s t x3 y z m 3 5 n m 1 5 n x3 2 y x3 4 y z 2 z 3 (w 2 z 1 4 ) 6 w 8 z 1 2 p9 q r1 5 s a1 3 b 1 6 c p 4 q 6 r x5 6 y 4 3 x 7 6 y p1 2 q 4 3 r p15 8 q 3 r In exercises 77 to 84, write each expression in radical form. Do not simplify. 77. a m x m x y (3x) (2y) 3 4 In exercises 85 to 88, write each expression using rational exponents, and simplify when necessary a w m 6 n r 10 s
10 ANSWERS In exercises 89 to 92, evaluate each expression, using a calculator. Round each answer to three decimal places Describe the difference between x 2 and x Some rational exponents, like, can easily be rewritten as terminating decimals 2 1 (0.5). Others, like, cannot. What is it that determines which rational numbers can 3 be rewritten as terminating decimals? In exercises 95 to 104, apply the appropriate multiplication patterns. Then simplify your result. 95. a 1 2 (a 3 2 a 3 4 ) 96. 2x 1 4 (3x 3 4 5x 1 4 ) 97. (a 1 2 2)(a 1 2 2) 98. (w 1 3 3)(w 1 3 3) 99. (m 1 2 n 1 2 )(m 1 2 n 1 2 ) 100. (x 1 3 y 1 3 )(x 1 3 y 1 3 ) 101. (x 1 2 2) (a 1 3 3) (r 1 2 s 1 2 ) (p 1 2 q 1 2 ) As is suggested by several of the preceding exercises, certain expressions containing rational exponents are factorable. For instance, to factor x 2 3 x 1 3 6, let u x 1 3. Note that x 2 3 (x 1 3 ) 2 u 2. Substituting, we have u 2 u 6, and factoring yields (u 3)(u 2) or (x 1 3 3)(x 1 3 2). In exercises 105 to 110, use this technique to factor each expression x 2 3 4x y 2 5 2y a 4 5 7a w 4 3 3w x x
11 ANSWERS In exercises 111 to 120, perform the indicated operations. Assume that n represents a positive integer and that the denominators are not zero x 3n x 2n 112. p 1 n p n (y 2 ) 2n 114. (a 3n ) 3 r n r n 117. (a 3 b 2 ) 2n 118. (c 4 d 2 ) 3m xn 2 x 1 2 n w n w n 3 bn b n In exercises 121 to 124, write each expression in exponent form, simplify, and give the result as a single radical x y a 31 3 w In exercises 125 to 130, simplify each expression. Write your answer in scientific notation ( ) ( ) ( ) ( ) ( ) ( ) While investigating rainfall runoff in a region of semiarid farmland, a researcher encounters the following formula: t C L xy Evaluate t when C 20, L 600, x 3, and y The average velocity of water in an open irrigation ditch is given by the formula V 1.5x2 3 y 1 2 z Evaluate V when x 27, y 16, and z Use the properties of exponents to decide what x should be to make each statement true. Explain your choices regarding which properties of exponents you decide to use. (a) (a 2 3 ) x a (c) a 2x a (b) (a 5 6 ) x a (d) (2a 2 3 ) x a
12 ANSWERS The geometric mean is used to measure average inflation rates or interest rates. If prices increased by 15% over 5 years, then the average annual rate of inflation is obtained by taking the 5th root of 1.15: (1.15) or ~2.8% The 1 is added to 0.15 because we are taking the original price and adding 15% of that price. We could write that as P 0.15P Factoring, we get P 0.15P P(1 0.15) P(1.15) In the introduction to this chapter, the following statement was made: From February 1990 through February 2000, the Bureau of Labor Statistics computed an inflation rate of 68.1%, which is equivalent to an annual growth rate of 2.51%. From February 1990 through February 2000 is 12 months. To what exponent was raised to obtain this average annual growth rate? 135. On your calculator, try evaluating ( 9) 4 2 in the following two ways: (a) (( 9) 4 ) 1 2 (b) (( 9) 1 2 ) 4 Discuss the results. Answers Not a real number x 35. y b x s 43. w x 47. a y a 4 b xy st pq x 2 5 y 1 2 z 61. a 1 2 b s 2 x r 4 y 2 mn s 3 2xz 3 2n r 2 t y 2 m xy a x x x (7a) m 2 n a 2 a a m n 101. x 4x r 2r 1 2 s 1 2 s 105. (x 1 3 1)(x 1 3 3) 107. (a 2 5 3)(a 2 5 4) 109. (x 2 3 2)(x 2 3 2) 111. x 5n 113. y 4n 115. r a 6n b 4n 119. x x y (a) 81; (b) not defined 646
Rational Exponents. Given that extension, suppose that. Squaring both sides of the equation yields. a 2 (4 1/2 ) 2 a 2 4 (1/2)(2) a a 2 4 (2)
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