III Problems 89-92, lise Ihe fol/owillg illformalion. H ",_ Find the period T of a pendulum whose length is 4 inches.

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1 74 CAPTER R Review n Problems 79-86, use a calcllawf to approximate each radical. Round YOlir answer to two decimal places. 79. v'2 HO.V7 H. V'4 H2. ~ 2 + V3 v'5-2 3V'5- v'2 O 2\13 - V'4 H3.3 _ v' v'2 + 4 H5. \13 "0. v'2 87. Calculating the Amount of Gasoline in a Tank An Exxon station stores its gasoline in underground tanks that are right circular cylinders l)ing on their sides. See the illustration. The volume V of gasoline in the tank (in gallons) is given by the formula v ~ 40h' h. where i is the height of the gasoline (in inches) as measured on a depth stick. (3) f = 12 inches, how many gallons of gasoline are in the tank? (b) f h = inch, how many gallons of gasoline are in the tank? 88. nclined Planes The final velocity v of an object in feel per second (ft/sec) after it slides down a frictionless inclined plane of height h feet is v ~ Y64h + v~ where Vo is the initial velocity (in ft/sec) of the object. v ~" (a) What is the final velocity v of an object that slides down a frictionless inclined plane of height 4 feet? Assume that the initial velocity is O. (b) What is the final velocity v of an object that slides down a frictionless inclined plane of height 16 feet'! Assume that the initial velocity is O. (e) What is the final velocity v of an object that slides down a frictionless inclined plane of height 2 feet with an initial velocity of 4 ft/sec? Problems 89-92, lise he fol/owillg illformalion. H ",_ Period of a Pendulum 'The period T. in seconds, of a pendulum of length, in feet, may be approximated using the formula T = 2"Y/32 n the following problems, express your answer both as a square root and as a decimal. Find the period T of a pendulum whose length is 64 feet. Find the period T of a pendulum whose length is 16 feet. Find the period T of a pendulum whose length is 8 inche& Find the period T of a pendulum whose length is 4 inches. Give an example it to explain why W ~ lal. to show that W is not equal to a. Use m RATONAL OBJECTVES EXPONENTS 2 Evaluate Expressions with Fractional Exponents Simplify Radicals Using Rational Exponents Our purpose in this section is to give a definition for"a raised to the power min," where a is a real number and min is a rational number. However, we want the definition to obey the laws of exponents slated for integer exponents in Seclion 4. For example, if the law of exponents (a')' = a" is to hold. then it must be true that, (3'/')' = 3'" = 3' = 3 That is, attn is a number that, when raised to the power n, is a. But this was the definition we gave of the principal nth root of a in Section 8. Thus, all' = va a 1 / 3 = va a 1 / 4 = \Yri

2 SECfON 9 Rational Exponents 75 and we can state the following definition: f a is a real number and 11 ~ 2 is an integer, then (1) provided that ~ exists. Writing Expressions Containing Fractional Exponents as Radicals 1 (a) 4'/ 2 = \1'4 = 2 (b) 8'/ 2 = v'8 = 2V2 (c) (_27)'/' = ~-27 = -3 (d) 16'/' = m = 2\J!2 - Note that if 11 is even and a < 0 then ~ and a'/n do not exist. We now seek a definition for am/fl. where m and n are integers containing no common factors (except and -) and 11 ~ 2. Again, we want the definition to obey the laws of exponents stated earlier. For example, amln = a",(l/n) = (am)1/n and am/n = u(t/n)m = (al/n)m f a is a real number and m and 11 are integers containing no common factors, with n ~ 2, then (2) provided that ~ exists. We have two comments about equation (2): 1. The exponent min must be in lowest terms and 11 must be positive. 2. [n simplifying the rational expression a mln, either ~ or (~r may be used, the choice depending on which is easier to simplify. Generally, taking the root first, as in (~r, is easier. Using Equation (2) (a) 4'/ 2 = (\1'4)' = 2' = 8 (b) (_8)4/' = (~)' = (_2)4 = 16 (c) (32f21 5 = (V'Tf2 = T2 =.!. 4 (d) 4 6/ ' = 4'/ 2 = (\1'4)' = 2' = 8 NOW WORK PROBLEM 7. - Based on the definition of a mln, no meaning is given to a mln if a is a negative real number and n is an even integer. The definitions in equations (1) and (2) were stated so that the Laws of Exponents would remain true for rational exponents. For convenience, we list again the Law of Exponents.

3 76 CHAPTER R Review Laws of Exponents f a and b are real numbers and rand s are rational numbers, then a- r = ā' (a'y = a" (~)' = :: (ab)' = a'. b' a' - = a'-s a' where it is assumed that all expressions used are defined. 2 Rational exponents can sometimes be used to simplify radicals. Simplifying Radicals Using Rational Exponents Solution Simplify each expression. (a) (~)2 (b) ~ (c) V'4 v'2 (a) (~)2 = (7 1 / 4 )2 = = 7' = Y7 (b) ~ = X 3 / 9 = X / 3 = -ojx (c) V'4 v'2 = 41/3. 21/2 = (22)1/3. 21/2 = 2 2 / 3 '2 1 / 2 = 2 7 / 6 = 2,2 1 / 6 = 2f!2 NOW WORK PROBLtM 17. The next example illustrates the use of the Laws of Exponents to simplify. Simplifying Expressions Containing Rational Exponents Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. 2xl/3 )-3 (9x2 yl/3 ) 1/2 (a) (x 2 / 3 Y)(X- 2 y)/2 (b) ( 2/3 (c) 1/3 Y X y Solution (a) (x2/3y)(x-2y)/2 = (x 2 / 3 Y)[(X- 2 )'!2y'/2] = X2/3yx-1 yl/2 = (x2/3 [)(y. y'/2) = x- /'y'/2 l/2 = X/3 2X/3 )-3 (y'/3)3 (y2/')' y' y' ( (b) y2/3 f 2X/3 = (2xl/'? = 2'(X/3? = 8x Equation (8), page 32 (c) (9X 2 yl/3)1/2 = (9x 2 -(/'»)/2 = xl/'y y'-(/') NOW WORK PROBllM 43.

4 SECTON 9 Rational Exponents 77 ~ The next two examples illustrate some algebra that you will need to know for certain calculus problems. Writing an Expression as a Single Quotient Write the following expression as a single quotient in which only positive exponents appear. (x' + f/' + x..!.(x' + t'/'. 2x 2 Solution 1 x' (x' + 1)'/' + x... (x' + t'/'. 2'x = (x' + f/' + )'/' ~ (x' + (x' + )'/'(x' (x'+)+x' = (x' + f/' (x' + 1)'/' + 1)1/' + x' NOW WORK PROBLEM 51. 2x' + = (x' + f' Factoring an Expression Containing Rational Exponents Sol uti 0 n Factor: 4x'/3(2x + ) + 2x 4 /3 We begin by looking for factors that are common to the two terms. Notice that 2 and x'/3 are common factors. Thus. 4x l /3(2x + 1) + 2x 4 /3 = 2x l /3[2(2x + ) + xl = 2x'/3(5x + 2) 9 EXERCSES n Problems 1-34, simplify each expression / /' 3. (-27)2/ 3 4. (_64)2/3 (~r /' 6. (-8t'/ / '1' e7/3 11. W-3/ (8 f cr' 16. (it 17. (V3)' 18. (\0/4)' 19. (v'5r' 20. (~r' /'. 3'/ / (3 71/3 6 5 / /3 4 1 / / '/ / 3. 3'/3 27. ~.@ 28. V'2. \0/4 29. (~r' 30. (~r' 31. (V'6)' 32. (V'S)3 33. V2 V'2 34. v'5~ l ----

5 78 CHAPTER R Review n Problems 354, simplify each expression. Express your answer so tllat only positive exponents occur. Assume that any variables are positive. 35. \o/? 39. X3/2X- / (X ' y)/3(xl)'/3 V1+X - x. 1 2V1+X + x (x + 4)1/' - 2x(x + 4t l /' x+4 36.~ 40. X S / 4 X- / (xy)/'(x'l)/' ( X/')'(~) /3 Y X 53. V4x vx::5. x > 5 2vx::5 5V4x + 3' x> - x> V?~ 41. (X'y')'/3 45. (16x'y-/')3/' ~v'x 42. (x'y')'/' 46. (4x-'yl/')'' n Problems 49-62, write each expression as a single quotient in which only positive exponents and/or radicals appear. x 1/2 1 + X 1/2 49. ( + X)/2 + 2(1 + x), x> - SO. 2xl/' + x, x> x(x' + 1)'/' + x'. ~x' + 1)-'/" 2x 52. (x + 1)'/3 + X'} (x + t'/3. x ~ x, ---- (x' - 1)1/' _ (x' - 1)'/' x' x' xv'X 61. 2v'X ( + x')' x>o x < -1 or x > \Y8x+1 ~x :-=== + -"'-"===. x ~ 2. x ~ - 3~(x - 2)' 24~(8x + )' v?"+1- x. 2x 2v?"+1 x (9 - X')/' + x'(9 - x'rl /' 58., -3 < x < 3 9-x 62. (x' + 4)1/' - x'(x' + 4r l /2 Xl +.; 2x( 1 - X,)'/ x3( - x'r'/3 3 ( _ x')'/' n Problems 63-72, factor each expression. Your answer should contain only positive exponents. 63. (x + 1)3/2 + x. ~(x + 1)1/', x", X / 2 (X 2 + x) - SX 3 / 2-8xl/ 2, X ~ (x' + 4t 3 + X. 4(x' + 4)'/'. 2x 69. 4(3x + 5)1/'(2x + 3)3/' + 3(3x + 5)'/'(2x + 3)1/'. x"'-l 70. 6(6x + 1)1/'(4x - 3)3/' + 6(6x + 1)'/3(4x - 3)1/'. x'" 64. (x' + 4)'/3 + x. ~(x' + 4)1/3. 2x 66. 6x l /'(2x + 3) + x 3 /'. 8. x '" x(3x + 4)'/3 + x'. 4(3x + 4)'/' 71.3['/' + ~ x'/'. x > 0 CHAPTER REVEW ;' Things To Know Classilicalion of numhers (p. 2) Counting Whole numbers numbers ntegers Rational numbers rrational numbers Real numbers Quotients of two integers (denominator not equal to 0); terminating or repeating decimals Nonrepeating decimals Rational or irrational numbers