Powers and Roots. 20 Sail area 810 ft 2. Sail area-displacement ratio (r) Displacement (thousands of pounds)

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1 C H A P T E R Powers and Roots Sail area-displacement ratio (r) Sail area 1 ft 4 6 Displacement (thousands of pounds) ailing the very word conjures up images of warm summer S breezes, sparkling blue water, and white billowing sails. But to boat builders, sailing is a serious business. Yacht designers know that the ocean is a dangerous and unforgiving place. It is their job to build boats that are not only fast, comfortable, and fun, but capable of withstanding the punishment inflicted by the wind and waves. Designing sailboats is a technical balancing act. A good boat has just the right combination of length, width (or beam), sail area, and displacement. A boat can always be made faster by increasing the sail area, but too much sail area increases the chance of capsize. Making the boat wider decreases the chance of capsize, but causes more resistance from the water and slows down the boat. There are four ratios commonly used to measure performance and safety for a yacht: ballast-displacement ratio, displacementlength ratio, sail area-displacement ratio, and capsize screening value. The formulas for these ratios involve powers and roots, which is the subject of this chapter. In Exercise 71 of Section.5 you will find the sail area-displacement ratio for a sailboat.

2 414 (-) Chapter Powers and Roots.1 ROOTS, RADICALS, AND RULES In this section Fundamentals Roots and Variables Product Rule for Radicals Quotient Rule for Radicals In Section 4.6 you learned the basic facts about powers. In this section you will study roots and see how powers and roots are related. Fundamentals We use the idea of roots to reverse powers. Because 9 and () 9, both and are square roots of 9. Because 4 16 and () 4 16, both and are fourth roots of 16. Because and (), there is only one real cube root of and only one real cube root of. The cube root of is and the cube root of is. nth Roots If a b n for a positive integer n, then b is an nth root of a. If a b, then b is a square root of a. If a b, then b is the cube root of a. If n is a positive even integer and a is positive, then there are two real nth roots of a. We call these roots even roots. The positive even root of a positive number is called the principal root. The principal square root of 9 is and the principal fourth root of 16 is and these roots are even roots. If n is a positive odd integer and a is any real number, there is only one real nth root of a. We call that root an odd root. Because 5, the fifth root of is and is an odd root. We use the radical symbol to signify roots. n a If n is a positive even integer and a is positive, then n a denotes the principal nth root of a. If n is a positive odd integer, then n a denotes the nth root of a. If n is any positive integer, then n. We read n a as the nth root of a. In the notation n a, n is the index of the radical and a is the radicand. For square roots the index is omitted, and we simply write a. E X A M P L E 1 Evaluating radical expressions Find the following roots: a) 5 b) 7 c) 6 64 d) 4 a) Because 5 5, 5 5. b) Because () 7, 7. c) Because 6 64, d) Because 4, 4 (4).

3 .1 Roots, Radicals, and Rules (-) 415 We can use the radical symbol to find a square root on a graphing, but for other roots we use the xth root symbol as shown. The xth root symbol is in the MATH menu. CAUTION In radical notation, 4 represents the principal square root of 4, so 4. Note that is also a square root of 4, but 4. Note that even roots of negative numbers are omitted from the definition of nth roots because even powers of real numbers are never negative. So no real number can be an even root of a negative number. Expressions such as 9, 4 1, and 6 64 are not real numbers. Square roots of negative numbers will be discussed in Section 9.5 when we discuss the imaginary numbers. Roots and Variables Consider the result of squaring a power of x: (x 1 ) x, (x ) x 4, (x ) x 6, and (x 4 ) x. When a power of x is squared, the exponent is multiplied by. So any even power of x is a perfect square. Perfect Squares The following expressions are perfect squares: x, x 4, x 6, x, x 1, x 1,... A can provide numerical support for this discussion of roots. Note that () not because x x when x is negative. Note also that the will not evaluate because 9. Since taking a square root reverses the operation of squaring, the square root of an even power of x is found by dividing the exponent by. Provided x is nonnegative (see Caution below), we have: x x 1 x, x 4 x, x 6 x, and x x 4. CAUTION If x is negative, equations like x x and x 6 x are not correct because the radical represents the nonnegative square root but x and x are negative. That is why we assume x is nonnegative. If a power of x is cubed, the exponent is multiplied by : (x 1 ) x, (x ) x 6, (x ) x 9, and (x 4 ) x 1. So if the exponent is a multiple of, we have a perfect cube. Perfect Cubes The following expressions are perfect cubes: x, x 6, x 9, x 1, x 15,... Since the cube root reverses the operation of cubing, the cube root of any of these perfect cubes is found by dividing the exponent by : x x 1 x, x 6 x, x 9 x, and x 1 x 4. If the exponent is divisible by 4, we have a perfect fourth power, and so on.

4 416 (-4) Chapter Powers and Roots E X A M P L E You can illustrate the product rule for radicals with a. Roots of exponential expressions Find each root. Assume that all variables represent nonnegative real numbers. a) x b) t 1 c) 5 s a) x x 11 because (x 11 ) x. b) t 1 t 6 because (t 6 ) t 1. c) 5 s s 6 because one-fifth of is 6. Product Rule for Radicals Consider the expression. If we square this product, we get ( ) () () Power of a product rule () and () 6. The number 6 is the unique positive number whose square is 6. Because we squared and obtained 6, we must have 6. This example illustrates the product rule for radicals. Product Rule for Radicals The nth root of a product is equal to the product of the nth roots. In symbols, n ab n a n b, provided all of these roots are real numbers. E X A M P L E You can illustrate the quotient rule for radicals with a. Using the product rule for radicals Simplify each radical. Assume that all variables represent positive real numbers. a) 4y b) y a) 4y 4 y Product rule for radicals y b) y y Simplify. y 4 y y 4 Product rule for radicals y 4 A radical is usually written last in a product. Quotient Rule for Radicals Because 6, we have 6, or 6 6. This example illustrates the quotient rule for radicals.

5 .1 Roots, Radicals, and Rules (-5) 417 Quotient Rule for Radicals The nth root of a quotient is equal to the quotient of the nth roots. In symbols, n n a b a n, b provided that all of these roots are real numbers and b. In the next example we use the quotient rule to simplify radical expressions. E X A M P L E 4 Using the quotient rule for radicals Simplify each radical. Assume that all variables represent positive real numbers. 1 t a) b) 9 x 6 y t a) 9 t 9 t Quotient rule for radicals 1 b) x 6 y x 1 y x y 6 7 Quotient rule for radicals M A T H A T W O R K 1 Blast off! Joseph Bursavich watches each Space Shuttle mission with particular attention. He is a Senior Computer Systems Designer for Martin Marietta, writing programs that support the building of the external tank that is the structural backbone of the Space Shuttle for NASA. Each tank, made up of three major parts, is 157 feet long and takes almost two years to build. To date, 74 tanks have been completed, and each has done its job in carrying cargo into space. Currently, Mr. Bursavich supports a team whose objective is to develop a superlightweight tank made of a mixture of aluminum and other metals. Reducing the weight of the tank is vital to the space station program because a lighter tank means that each mission can carry a greater payload into space. Because of economic considerations, as many as ten external tanks might be produced at one time. Mr. Bursavich writes programs to determine the economic order quantity (EOQ) for components of the external tank. The EOQ depends on setup costs, labor costs, the quantity of the component to be used in one year, the cost of holding stock for one year, and maintenance costs. In Exercise of this section you will use the formula that Mr. Bursavich uses to determine the EOQ for component parts of the tanks. COMPUTER SYSTEMS DESIGNER

6 41 (-6) Chapter Powers and Roots WARM-UPS True or false? Explain your answer. 1. True. False. 7 True False True 6. 9 True 7. 9 False False 9. If w, then w w. True 1. If t, then 4 t 1 t. True. 1 EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. How do you know if b is an nth root of a? If b n a, then b is an nth root of a.. What is a principal root? The principal root is the positive even root of a positive number.. What is the difference between an even root and an odd root? If b n a, then b is an even root provided n is even or an odd root provided n is odd. 4. What symbol is used to indicate an nth root? The nth root of a is written as n a. 5. What is the product rule for radicals? The product rule for radicals says that n a n b n ab provided all of these roots are real. 6. What is the quotient rule for radicals? The quotient rule for radicals says that n a n b n ab provided all of these roots are real. For all of the exercises in this section assume that all variables represent positive real numbers. Find each root. See Example Not a real number Not a real number Not a real number Not a real number Not a real number. 144 Not a real number Find each root. See Example. 1. m m. m 6 m. 5 y 15 y 4. 4 m m 5. y 15 y 5 6. m m 4 7. m m. 4 x 4 x ,, Use the product rule for radicals to simplify each expression. See Example y y 4. 16n 4n 49. 4a a 5. 6n 6n 51. xy 4 x y 5. w 6 t w t 5. 5m 1 m z 16 z y y 56. 7z z 57. 7w w 5. 15m 6 5m s 4 s w 4 w a 9 y 6 5a y 6. 7zw 15 zw 5 Simplify each radical. See Example t 4 t 64. w t t 69. y x6 x y 6. a y 6 w a y 1 1

7 .1 Roots, Radicals, and Rules (-7) a y a 7. 9a 4 9 b 4 a 7 b 6 4 y w 1 4 Use a to find the approximate value of each expression to three decimal places (1.)( ) ()(.).56 4(1)(.9) (). 4(1.)(6.) (1.) 6.5 Solve each problem.. Economic order quantity. When a part is needed for a space shuttle external fuel tank, Joseph Bursavich at Martin Marietta determines the most economic order quantity E by using the formula E A I S, where A is the quantity that the plant will use in one year, S is the cost of setup for making the part, and I is the cost of holding one unit in stock for one year. Find the most economic order quantity if S $59, A, and I $ Diagonal of a box. The length of the diagonal D of the box shown in the figure can be found from the formula D L W H, where L, W, and H represent the length, width, and height of the box, respectively. If the box has length 6 inches, width 4 inches, and height inches, then what is the length of the diagonal to the nearest tenth of an inch? 7. in. in. 4 in. 5w FIGURE FOR EXERCISE 4 5. Buena vista. The formula V 1.A gives the view in miles from horizon to horizon at an altitude of A feet (Delta Airlines brochure). D 6 in. View (miles) a) Use the formula to find the view to the nearest mile from an altitude of 5, feet. b) Use the accompanying graph to estimate the altitude of an airplane from which the view is 1 miles. a) mi b) 7 ft 1 Maximum speed (knots) Altitude (thousands of feet) FIGURE FOR EXERCISE 5 6. Sailing speed. To find the maximum speed in knots (nautical miles per hour) for a sailboat, sailors use the formula M 1.w, where w is the length of the waterline in feet. a) If the length of the waterline for the sloop John B. is feet, then what is the maximum speed for the John B.? b) Use the accompanying graph to estimate the length of the waterline for a boat for which the maximum speed is 6 knots. a). knots b) ft Waterline (feet) FIGURE FOR EXERCISE 6 GETTING MORE INVOLVED 7. Discussion. Determine whether each equation is correct. a) (5) 5 b) () c) 4 () 4 d) 5 (7) 5 7 a) no b) yes c) no d) yes. Writing. If x is a negative number and n x n x, then what can you say about n? Explain your answer. n is odd