RATIONAL EXPONENTS 9.2. section. calculator. close-up. Rational Exponents. Solution. Solution

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1 78 (9 ) Chpter 9 Rdicls nd Rtionl Exponents In this Rtionl Exponents section Using the Rules of Exponents Simplifying Expressions Involving Vribles clcultor close-up You cn find the fifth root of using rdicl nottion or exponent nottion. Note tht the frctionl exponent 5 must be in prentheses. 9. RATIONAL EXPONENTS You hve lerned how to use exponents to express powers of numbers nd rdicls to express roots. In this section you will see tht roots cn be expressed with exponents lso. The dvntge of using exponents to express roots is tht the rules of exponents cn be pplied to the expressions. Rtionl Exponents The nth root of number cn be expressed by using rdicl nottion or the exponent n. For exmple, 8 nd 8 both represent the cube root of 8, nd we hve 8 8. Definition of /n If n is ny positive integer, then n n, provided tht n is rel number. Lter in this section we will see tht using exponent n for nth root is comptible with the rules for integrl exponents tht we lredy know. E X A M P L E Rdicls or exponents Write ech rdicl expression using exponent nottion nd ech exponentil expression using rdicl nottion. ) 5 b) xy c) 5 d) 5 ) 5 5 b) xy (xy) c) 5 5 d) 5 5 In the next exmple we evlute some exponentil expressions. E X A M P L E Finding roots Evlute ech expression. ) b) (8) c) 8 d) (9) ) b) (8) 8 c) 8 8 d) Becuse (9) or 9 is n even root of negtive number, it is not rel number.

2 9. Rtionl Exponents (9 ) 79 We now extend the definition of exponent n to include ny rtionl number s n exponent. The numertor of the rtionl number indictes the power, nd the denomintor indictes the root. For exmple, the expression Power 8 Root represents the squre of the cube root of 8. So we hve 8 (8 ) (). helpful hint Note tht in mn we do not require mn to be reduced. As long s the nth root of is rel, then the vlue of mn is the sme whether or not mn is in lowest terms. Definition of m/n If m nd n re positive integers, then mn ( n ) m, provided tht n is rel number. We define negtive rtionl exponents just like negtive integrl exponents. Definition of m/n If m nd n re positive integers nd 0, then mn m, n provided tht n is rel number. E X A M P L E Rdicls or exponents Write ech rdicl expression using exponent nottion nd ech exponentil expression using rdicl nottion. ) x b) m c) 5 d) 5 ) x x b) m m m c) 5 5 d) 5 5 To evlute n expression with negtive rtionl exponent, remember tht the denomintor indictes root, the numertor indictes power, nd the negtive sign indictes reciprocl: mn Root Power Reciprocl

3 80 (9 ) Chpter 9 Rdicls nd Rtionl Exponents The root, power, nd reciprocl cn be evluted in ny order. However, to evlute mn mentlly it is usully simplest to use the following strtegy. Strtegy for Evluting m /n Mentlly. Find the nth root of.. Rise your result to the mth power.. Find the reciprocl. For exmple, to evlute 8 mentlly, we find the cube root of 8 (which is ), squre to get, then find the reciprocl of to get. In print 8 could be written for evlution s ((8 ) ) or (8. ) E X A M P L E clcultor close-up A negtive frctionl exponent indictes reciprocl, root,nd power. To find you cn find the reciprocl first, the squre root first, or the third power first s shown here. Rtionl exponents Evlute ech expression. ) 7 b) c) 8 d) (8) 5 ) Becuse the exponent is, we find the cube root of 7 nd then squre it: 7 (7 ) 9 b) Becuse the exponent is, we find the squre root of, cube it, nd find the reciprocl: ( ) 8 c) Becuse the exponent is, we find the fourth root of 8, cube it, nd find the reciprocl: 8 (8 ) Definition of negtive exponent 7 d) (8) 5 ((8 ) ) 5 ( ) 5 CAUTION An expression with negtive bse nd negtive exponent cn hve positive or negtive vlue. For exmple, (8) 5 nd (8). Using the Rules of Exponents All of the rules for exponents hold for rtionl exponents s well s integrl exponents. Of course, we cnnot pply the rules of exponents to expressions tht re not rel numbers.

4 9. Rtionl Exponents (9 5) 8 Rules for Rtionl Exponents The following rules hold for ny nonzero rel numbers nd b nd rtionl numbers r nd s for which the expressions represent rel numbers.. r s rs Product rule. r s rs Quotient rule. ( r ) s rs Power of power rule. (b) r r b r Power of product rule 5. b r b r r Power of quotient rule We cn use the product rule to dd rtionl exponents. For exmple, The fourth root of 6 is, nd squred is. So 6. Becuse we lso hve 6, we see tht rtionl exponent cn be reduced to its lowest terms. If n exponent cn be reduced, it is usully simpler to reduce the exponent before we evlute the expression. We cn simplify 6 6 s follows: E X A M P L E 5 Using the product nd quotient rules with rtionl exponents Simplify ech expression. ) b) 5 5 ) Product rule for exponents 7 9 b) We used the quotient rule to subtrct the exponents. E X A M P L E 6 Using the power rules with rtionl exponents Simplify ech expression. 6 ) b) ( 0 ) c) 9 ) Becuse the bses nd re different, we cnnot use the product rule to dd the exponents. Insted, we use the power of product rule to plce the power outside the prentheses: ( ) 6 6

5 8 (9 6) Chpter 9 Rdicls nd Rtionl Exponents b) Use the power of power rule to multiply the exponents: ( 0 ) ( ) c) 9 9 ( ) Power of quotient rule Power of power rule 7 Definition of negtive exponent helpful hint We usully think of squring nd tking squre root s inverse opertions, which they re s long s we stick to positive numbers.we cn squre to get 9, nd then find the squre root of 9 to get wht we strted with. We don t get bck to where we begn if we strt with. Simplifying Expressions Involving Vribles When simplifying expressions involving rtionl exponents nd vribles, we must be creful to write equivlent expressions. For exmple, in the eqution (x ) x it looks s if we re correctly pplying the power of power rule. However, this sttement is flse if x is negtive becuse the power on the left-hnd side indictes the positive squre root of x. For exmple, if x, we get [() ] 9, which is not equl to. To write simpler equivlent expression for (x ), we use bsolute vlue s follows. Squre Root of x (x ) x for ny rel number x. Note tht (x ) x is lso written s x x. Both of these equtions re identities. It is lso necessry to use bsolute vlue when writing identities for other even roots of expressions involving vribles. E X A M P L E 7 Using bsolute vlue symbols with roots Simplify ech expression. Assume the vribles represent ny rel numbers nd use bsolute vlue symbols s necessry. 9 ) (x 8 y ) b) x 8 ) Apply the power of product rule to get the eqution (x 8 y ) x y. The lefthnd side is nonnegtive for ny choices of x nd y, but the right-hnd side is negtive when y is negtive. So for ny rel vlues of x nd y we hve (x 8 y ) x y.

6 9. Rtionl Exponents (9 7) 8 b) Using the power of quotient rule, we get 9 x x 8. This eqution is vlid for every rel number x, so no bsolute vlue signs re used. Becuse there re no rel even roots of negtive numbers, the expressions, x, nd y 6 re not rel numbers if the vribles hve negtive vlues. To simplify mtters, we sometimes ssume the vribles represent only positive numbers when we re working with expressions involving vribles with rtionl exponents. Tht wy we do not hve to be concerned with undefined expressions nd bsolute vlue. E X A M P L E 8 Expressions involving vribles with rtionl exponents Use the rules of exponents to simplify the following. Write your nswers with positive exponents. Assume ll vribles represent positive rel numbers. ) x x b) c) (x y ) d) x y ) x x x 6 Use the product rule to dd the exponents. x Reduce the exponent. b) Simplify. c) (x y ) (x ) (y ) Power of product rule x y Power of power rule Use the quotient rule to subtrct the exponents. x Definition of negtive exponent y d) Becuse this expression is negtive power of quotient, we cn first find the reciprocl of the quotient, then pply the power of power rule: x y y y x 6 x 6 WARM-UPS True or flse? Explin your nswer (6) ( 8 ) 6

7 8 (9 8) Chpter 9 Rdicls nd Rtionl Exponents 9. EXERCISES Reding nd Writing After reding this section, write out the nswers to these questions. Use complete sentences.. How do we indicte n nth root using exponents?. How do we indicte the mth power of the nth root using exponents?. Wht is the mening of negtive rtionl exponent?. Which rules of exponents hold for rtionl exponents? 5. In wht order must you perform the opertions indicted by negtive rtionl exponent? 6. When is mn rel number? Write ech rdicl expression using exponent nottion nd ech exponentil expression using rdicl nottion. See Exmple cbs x. y.. (b) 5 Evlute ech expression. See Exmple (5) 8. () (). (6) Write ech rdicl expression using exponent nottion nd ech exponentil expression using rdicl nottion. See Exmple.. w w (b) 0. (m) 5 Evlute ech expression. See Exmple (7) 0. (8). (6). (00) Use the rules of exponents to simplify ech expression. See Exmples 5 nd ( 6 ) 5. ( 0 ) ( 8 ) 56. ( 6 ) 57. ( ) 58. (5 ) Simplify ech expression. Assume the vribles represent ny rel numbers nd use bsolute vlue s necessry. See Exmple (x ) 6. (y 6 ) 6 6. ( 8 ) 6. (b 0 ) 65. (y ) 66. (w 9 ) 67. (9x 6 y ) 68. (6 8 b ) x 0 y y Simplify. Assume ll vribles represent positive numbers. Write nswers with positive exponents only. See Exmple x x 7. y y 7. (x y)(x y ) 7. ( b )(b) 75. w 76. w 77. (x 6 ) 78. (5 8 ) 79. b 80. b 6 Simplify ech expression. Write your nswers with positive exponents. Assume tht ll vribles represent positive rel numbers. 8. (9 ) 8. ( 6 )

8 9. Rtionl Exponents (9 9) (9x 9 ) 98. (7x 9 ) 99. ( ) 00. (5x ) 0. ( b) (b ) 0. (m n ) (m n ) 0. (km ) (k m 5 ) 0. (tv ) (t v ) Use scientific clcultor with power key (x y ) to find the deciml vlue of ech expression. Round nswers to four deciml plces In Exercises 0, solve ech problem.. Digonl of box. The length of the digonl of box cn be found from the formul D (L W H ), where L, W, nd H represent the length, width, nd height of the box, respectively. If the box is inches long, inches wide, nd inches high, then wht is the length of the digonl? in. FIGURE FOR EXERCISE. Rdius of sphere. The rdius of sphere is function of its volume, given by the formul r 0.7 5V. Find the rdius of sphericl tnk tht hs volume cubic meters. of in. D in () r. 6 5, Simplify ech expression. Assume nd b re positive rel numbers nd m nd n re rtionl numbers. 5. m m 6. b n b n 7. m5 m 8. b b n n 9. ( m b n ) mn 0. ( m b n ) 6 mb. 9m 6n. m 6 n b 6m 9 n b FIGURE FOR EXERCISE 5. Mximum sil re. According to the new Interntionl Americ s Cup Clss Rules, the mximum sil re in squre meters for ycht in the Americ s Cup rce is given by S ( D 0.8L), where D is the displcement in cubic meters (m ), nd L is the length in meters (m). (Scientific Americn, My 99). Find the mximum sil re for bot tht hs displcement of 8. m nd length of.5 m.

9 86 (9 0) Chpter 9 Rdicls nd Rtionl Exponents L (m) S (m ) D (m ) FIGURE FOR EXERCISE 5 6. Orbits of the plnets. According to Kepler s third lw of plnetry motion, the verge rdius R of the orbit of plnet round the sun is determined by R T, where T is the number of yers for one orbit nd R is mesured in stronomicl units or AUs (Windows to the Universe, ) It tkes Mrs.88 yers to mke one orbit of the sun. Wht is the verge rdius (in AUs) of the orbit of Mrs? b) The verge rdius of the orbit of Sturn is 9.05AU. Use the ccompnying grph to estimte the number of yers it tkes Sturn to mke one orbit of the sun. 8. Best bond fund. The top bond fund for 997 in the 5-yer ctegory ws GT Globl High Income B. An investment of $0,000 in 99 grew to $,80.95 in 997. Use the formul from the previous exercise to find the 5-yer verge nnul return for this fund. 9. Overdue lon pyment. In 777 welthy Pennsylvni merchnt, Jcob DeHven, lent $50,000 to the Continentl Congress to rescue the troops t Vlley Forge. The lon ws not repid. In 990 DeHven s descendnts filed suit for $.6 billion (New York Times, My 7, 990). Wht verge nnul rte of return were they using to clculte the vlue of the debt fter yers? (See Exercise 7.) 0. Cliforni growin. The popultion of Cliforni grew from 9.9 million in 970 to.5 million in 000 (U.S. Census Bureu, Find the verge nnul rte of growth for tht time period. (Use the formul from Exercise 7 with P being the initil popultion nd S being the popultion n yers lter.) 5 Rdius of orbit (AU) R T / Time for one orbit (yers) FIGURE FOR EXERCISE 6 7. Best stock fund. The verge nnul return for n investment is given by the formul r S P n, where P is the initil investment nd S is the mount it is worth fter n yers. The top mutul fund for 997 in the -yer ctegory ws Fidelity Select-Energy Services (Money Guide to Mutul Funds, 998), in which n investment of $0,000 grew to $, from 99 to 997. Find the -yer verge nnul return for this fund. Cliforni popultion (millions of people) Yer FIGURE FOR EXERCISE 0 GETTING MORE INVOLVED. Discussion. If we use the product rule to simplify () (), we get () () (). If we use the power of product rule, we get () () ( ). Which of these computtions is incorrect? Explin your nswer.. Discussion. Determine whether ech eqution is n identity. Explin. ) (w x ) w x b) (w x ) wx c) (w x ) w x