CHAPTER 8 ROOTS AND RADICALS

Chapter Eight Additional Exercises 09 CHAPTER 8 ROOTS AND RADICALS Section 8.1 Evaluating Roots Objective 1 Find square roots. Find all square roots of the number. 1. 81. 19.. 00... 1 8. 1 9. 9 10. 11 1 11. 19 1. 19 89 Find the square root. 1. 100 1. 1 1. 1 1. 1. 18. 900 19. 10, 000 0. 9 1.. 19. 1 1. 00 00 Objective Decide whether a given root is rational, irrational, or not a real number. Tell whether the square root is rational, irrational, or not a real number.. 1.. 1 8. 9. 9 0. 1. 100. 00. 9.... 0.9 8

10 Chapter Eight Additional Exercises Objective Find decimal approximations for irrational square roots. Use a calculator to find a decimal approximation for the root. Round answers to the nearest thousandth.. 8. 1 9. 1 0. 90 1. 10. 00. 11.. 8. 1100. 1, 00 8., 8 Objective Use the Pythagorean formula. Find the length of the unknown side of the right triangle with sides a, b, and c, where c is the hypotenuse. If necessary, round your answer to the nearest thousandth. 9. a = 8, b = 1 0. a = 8, b = 1 1. c =, a =. c =, b =. c = 1, a = 1. a =, b = 8 Use the Pythagorean formula to solve the problem. If necessary, round your answer to the nearest thousandth.. The hypotenuse of a right triangle measure 10 centimeters and one leg measures centimeters. How long is the other leg?. Two sides of a rectangle measure centimeters and 11 centimeters. How long are the diagonals of the rectangle?. A diagonal of a rectangle measures 9 inches. The length of the rectangle is 1 inches. Find the width of the rectangle. 8. A diagonal of a rectangle measures. meters. The width of the rectangle is 1. meters. Find the length of the rectangle.

Chapter Eight Additional Exercises 11 9. A ladder feet long leans against a wall. The foot of the ladder is feet from the base of the wall. How high on the wall does the top of the ladder rest? 0. Laura is flying a kite on 0 feet of string. How high is the kite above her hand (vertically) if the horizontal distance between Laura and the kite is feet? 1. Kevin started to drive due south at the same time Lydia started to drive due west. Lydia drove 1 miles in the same time that Kevin drove 8 miles. How far apart were they at that time?. A cable from the top of a pole 1 feet tall is pulled taut and attached to the ground 8 feet from the base of the pole. How long is the cable?

1 Chapter Eight Additional Exercises. The foot of a loading ramp 9 feet long is placed feet from the base of a platform. The top of the ramp rests on the platform. How high is the platform?. A plane flies due east for miles and then due south until it is miles from its starting point. How far south did the plane fly? Objective Use the distance formula. Find the distance between each pair of points.. (,),(,8). (,),( 9,1). (,),( 0, ) 8. (, ),(, ) 9. (, 1 ),(, ) 0. ( 0,0),(, ) 1. (, ),(, ). (,0),( 0, ). (, ),(,1). (, 8),(,1). (, ),(, ). (, 11 ),( 9, ) Objective Find cube, fourth, and other roots. Find each root that is a real number.. 8. 1 9. 1 80. 1000 81. 1 8. 8 8. 8. 8. 81 8. 8. 88. 89. 1 90. 100, 000 91. 1 9. 1

Chapter Eight Additional Exercises 1 Mixed Exercises Find each root that is a real number. 9. 19 9. 1 9. 1 9. 9. 98. 81 99. 100. 89 1 Write rational, irrational, or not a real number to describe the number. If the number is rational, give its exact value. If the number is irrational, give a decimal approximation to the nearest thousandth. Use a calculator as necessary. 101. 10. 10. 9 10. 10. 81 10. 19 1 1 10. 8. 108. 0.09 Find the length of the unknown side of the right triangle with sides a, b, and c, where c is the hypotenuse. If necessary, round your answer to the nearest thousandth. 109. a = 10, b = 110. b =, c = 9 111. a = 1, c = 1 11. c = 1, a = 0 11. a =, b = 8 11. b = 8, c = 11 11. a = 9, c = 1 11. b =, c = 11 Writing/Conceptual Exercises Answer the question. 11. How many fourth roots does 0 have? 118. How many real number fourth roots does any positive number have? 119. How many real number fourth roots does any negative number have?

1 Chapter Eight Additional Exercises 10. Which of the following numbers have rational square roots? (a) (b) 9 (c) 9 (d) 11 11. Which of the following numbers have irrational square roots? (a) (b) 1 (c) 1 (d) 9 What must be true about b for the statement to be true? 1. b is a negative number. 1. b is a negative number. 1. b is a positive number. 1. b is a positive number. 1. b is zero. Section 8. Multiplying, Dividing, and Simplifying Radicals Objective 1 Multiply square root radicals. Use the product rule for radicals to find the product. 1. 18. 1 19. 10. 11. 1. 1. 1. 0 1. 1 11 1. r, r > 0 1. x, x > 0 18. 18 a b, a > 0, b > 0

Chapter Eight Additional Exercises 1 Objective Simplify radicals by using the product rule. Simplify the radical. Assume that all variables represent nonnegative real numbers. 19. 10. 11. 00 1. 10 1. 1. 88 1. 1. 1000 1. y 18. p q 19. 80a 10. 1r s Find the product and simplify. 11. 1 1. 0 1. 1 1. 0 10 1. 11 1. 18 Objective Simplify radicals by using the quotient rule. Use the quotient rule and product rule, as necessary, to simplify the expression. 1. 19 11 18. 1 9 19. 10. 9 00 11. 8 1. 1. 11 1 1 1. 1. 18 8 Objective Simplify radicals involving variables. 1. 8 x 1. p 18. x y 19. a 0a 10. 11. 8x 1. 81x 1. x 8 1. x

1 Chapter Eight Additional Exercises Objective Simplify higher roots. Simplify the expression. 1. 1. 81 1. 18 18. 19. 180. 1 1 181. 18. 18. 1 81 18. 9 18. 1 18. Mixed Exercises Find the product and simplify if possible. 18. 11 11 188. 1 1 189. 190. 10 0 191. 1 19. 9 9 Simplify the expression. Assume that all variables represent positive numbers. 19. 99 19. 19. 1 19. 1 19. 8 198. 9x x 199. 1 00. 8 11 01. 0a 0. a 0. 0 0. 1 10 0. a b 0. 0 0. 08. 09. 98 y x 10.

Chapter Eight Additional Exercises 1 Writing/Conceptual Exercises Decide whether the statement is true or false. 11. = 1. + = + 1. ( 10) = 10 1. ( 10) = 10 1. Which of the following radicals are simplified according to the guidelines given in the textbook? (a) (b) 19 (c) 1 (d) 1. Which of the following radicals are simplified according to the guidelines given in the textbook? (a) 9 (b) 1 (c) 1 (d) 0 Section 8. Adding and Subtracting Radicals Objective 1 Add and subtract radicals. Add or subtract, as indicated. 1. + 9 18. 11 11 19. 10 + 10 0. 1. +. +. +. +. 9 1 + 1 1. + Objective Simplify radical sums and differences. Simplify and add or subtract terms wherever possible. Assume that all variables represent nonnegative real numbers.. 8 + 8. + 9. 00 18 0. 11

18 Chapter Eight Additional Exercises 1. + 0. 8 + 8 1 8. 0 1.. 8. 18 8 8. 8z 18z 8. 9w w 9. 0x 00x 0. 1 1. 8. r + r Objective Simplify more complicated radical expressions. Perform the indicated operations. Assume that all variables represent nonnegative real numbers.. + 1.. 1. 8 +. x + x 8. y 18 y 9. 10x 1x x 0. 11 w 0w 8w 1. k k + k k. y 80y Mixed Exercises Simplify and perform the indicated operations. Assume that all variables represent nonnegative real numbers.. 19 19. 11 + 11 11. + 1. 80 +. m m 8. 1

Chapter Eight Additional Exercises 19 9. 11 m 0. r 0 8r 1 1. 8 1. + 0 8 10. 18 + 8 8. 18 + 0. 1 1 +. 18 + 1. 1 + 0 8. 8 9. 8 0. 98 8 + 1. + 1 0. + 81. 8z + z + z. 81 + + 19. 8 y + 1y + y. 1r + r 1r. 8 + 8. 100 x 9x + x Writing/Conceptual Exercises 9. Write an equation showing how the distributive property is used to justify the statement + 8 = 1. 80. Write an equation showing how the distributive property is used to justify the statement 11 =. Despite the fact that and 8 are radicals that have different root indexes, they can be added to obtain a single term: + 8 = + =. 81. Make up a similar sum of radicals that leads to an answer of 1. 8. Make up a similar difference of radicals that leads to an answer of 8.

0 Chapter Eight Additional Exercises Section 8. Rationalizing the Denominator Objective 1 Rationalize denominators with square roots. Rationalize the denominator. Write all answers in simplest form. 8. 8. 1 8. 8. 1 8. 88. 10 89. 11 90. 8 1 91. 9. 8 9. 1 9. 8 9. 9. 1 9. 0 98. 9 Objective Write radicals in simplified form. Perform the indicated operations and write all answers in simplest form. Rationalize all denominators. Assume that all variables represent positive real numbers. 99. 1 00. 01. 8 0. 0. 0 0. 8 11 0. 0. 0. 1 0 1 08. y x 09. r s 10. a b 11. k k m df 1. d 1. 0a b a

Chapter Eight Additional Exercises 1 Objective Rationalize denominators with cube roots. Rationalize the denominator. Assume that all variables in the denominator represent nonzero real numbers. 1. 9 1. 9 1. 1 1 1. 18. 19. 8 0. 1. 1.. 80 1. 1. r s. 9 r. v w 8t 8. u Mixed Exercises Simplify the expression. Assume that all variables represent positive real numbers. 9. 0. 1... 11 0. 1.. 1. x 8. 1 qt 9. 98 t 0. x 81y 1. 1 r. 98. 8

Chapter Eight Additional Exercises. 1 t x.. y t 1x. t 8. 9. 8x 0. 19 1. c d. t Writing/Conceptual Exercises. Which one of the following would be an appropriate choice for multiplying the numerator and denominator of in order to rationalize the denominator? (a) (b) (c) 9 (d). Which one of the following would be an appropriate choice for multiplying the y numerator and denominator of in order to rationalize the denominator? 10z (a) 10z (b) 100z (c) 10z (d) 100z. Which one of the following would be an appropriate choice for multiplying the s numerator and denominator of in order to rationalize the denominator? 9r (a) r (b) 9r (c) r (d) s. What would be your first step in simplifying the radical? What property would 11 you be using in this step?

Chapter Eight Additional Exercises Section 8. More Simplifying and Operations with Radicals Objective 1 Simplify products of radical expressions. Simplify the expression.. ( ) 8. 10 ( + ) 9. ( 1 + ) 0. ( 11 ) 1. ( + )( + ). ( 11)( + ). ( )( + 10). ( )( + ). ( + 10)( 10). ( )( + ). ( ) 8. ( 10 11) 9. ( + ) 0. ( + ) Objective Use conjugates to rationalize denominators of radical expressions. Rationalize the denominator. 1.. +.. +. 1. +. 9 + 8. + 10 9. + 80. + 1 81. + 8. 10 + 10

Chapter Eight Additional Exercises Objective Write radical expressions with quotients in lowest terms. Write the quotient in lowest terms. 8. 8 8. + 1 8. + 1 8. + 8 1 8. 9 + 1 88. 10 89. + 8 90. + 91. 8 1 1 9. 1 + 8 9. 1 + 1 9. 1 Mixed Exercises Simplify the expression. 9. ( 11) 9. ( )( + 1) 9. ( 11 + ) 98. ( + 1)( 1) 99. 1 11 00. + 01. 1 + 1 0. 1 0. 0. + 0. ( 1 + )( 1 ) 0. ( + ) 0. ( 1) 08. ( 10 + )( 11) 09. ( )( ) + 10. 11 +

Chapter Eight Additional Exercises Write the quotient in lowest terms. 11. 8 1. 0 10 10 1. 10 9 1. 11 1 10 1. 1 1. 19 1 9 1. 98 1 18. 1 19. 1 1 0. 9 + 0 1 Section 8. Solving Equations with Radicals Objective 1 Solve radical equations having square root radicals. Solve the equation. 1. w = 9. q 1 = 0. 8 y =. b +1 =. q + =. z = 0. + k = k 8. r + 1 = r 9. x + = x 0. t + = 0 1. t + = t 1. x + 18 = x + 1

Chapter Eight Additional Exercises Objective Identify equations with no solutions. Solve the equation if it has a solution.. y = 10. 8 p = 0. z + = 0. 0 = r. y + = 1 8. k + = 0 9. x + + = 0 0. m + 1 = m + 1. n = n. p = p +. r = r r + 1. s = s + s +. b = b b + 1. d + d + 1 + d = 0 Objective Solve equations by squaring a binomial. Find all solutions for the equation.. y + 1 = y + 1 8. m + = m + 9. x + = x 0. x + 1 1 = x 1. t + = t + t +. q 1 = q + 8q + 11. p = p +. a 1 = a + 1. x = x 8. y + 1 = y + 1. b = b 8. p + 1 = p + 1 9. c + = c + 0. t = t + 1. x + 1 = x + 9. x x + = 0

Chapter Eight Additional Exercises. t + t = 1. a 1 + 1 = a +. k + = k + 1. x + 1 = + x. x + = x + + 1 8. r + r + 11 = Objective Solve radical equations having cube root radicals. Solve each equation. 9. x+ 1= x 0. x = x+ 1. x = x. x = x. x + 1x 8 = x. 8x + x = x Mixed Exercises Find all solutions for the equation.. z 8 =. q + =. r + = 0 8. p + = 0 9. r = 8r + 1 80. y = y + 1 81. x+ 1= x+ 1 8. x + 1 = x 8. n + = n 8. m + = 8. t = t t+ 8. t = 9t t + 1 8. x + = x 88. m + = m 1 89. p = p p 1 90. k k k = + 10

8 Chapter Eight Additional Exercises 91. x = x 10 9. r = r + r 0 9. x + 1 = x 1 9. x + 1 = x + 1 9. y = 0 9. x = x 9. y + 1 = y + 98. a + a + = 99. x 1 + = 0 00. c + = c + + 01. q 8 = q 0. k + 10 = k + 19 0. x+ = x 0. x = x 0. x = x 0. x = x 9 0. x + x = 10x 08. x + x = 1x Writing/Conceptual Exercises 09. How can you tell that the equation x + 1 = 9 has no real number solution without performing any algebraic steps? 10. Explain why the equation x = 81 has two real number solutions, while the equation x = 9 has only one real number solution. 11. A student is told that he must check his solutions to the equation r = r. The student said that he doesn t see why this is necessary because he is sure he didn t make any mistakes in solving the equation. How would you respond? 1. Explain why the equation z = z + z + cannot have a negative solution.

Chapter Eight Additional Exercises 9 Section 8. Using Rational Numbers as Exponents Objective 1 Define and use expressions of the form Evaluate the expression. n a 1 /. 1. 1/ 1 1. 1/ 1. 1/ 19 1. 1/ 1. 1/ 8 18. 1/ 19. 1/ 0. 1/ 1 1. 1/ 1. 1/ 1. 1/ 9. 1/ 1000. 1/ 1. 1/ 81. 1/ 8. 1/ 19 9. 1/ 0. 1/ 1. 1/. 1/ 18 Objective Define and use expressions of the form m n a /. Evaluate the expression.. / 1. /. /. /. / 8 8. / 8 9. / 0. / 1. /. / 1. / 1. / 1. / 8. /. / 8. / 9. / 1 0. / 1. /. 1000 / Objective Apply the rules for exponents using rational exponents. Simplify the expression. Write the answer in exponential form with only positive exponents. Assume that all variables represent positive numbers.. 1/ /. / /. / 8 / 8. 8 / /. 8 / 8 / 8 8 8. 9 1/ 9 / / 9. 1/ 8 1/ / 0. 1. 1/ / 8

0 Chapter Eight Additional Exercises. ( r s / ) /. ( 9 / ). ( / ) 10 8 1 /.. 1/ 1 9 /. / 8. a b / 1/ 1 9. r s 1/ 1/ / 1/ y y 0. y / 1. r r r / 1/ / 8 8. 8 8 / / z z. / z x. x 1/ / /. c x / 1 1/. ( ) 1/ ( ) a a Objective Use rational exponents to simplify radicals. Simplify the radical by first writing it in exponential form. Give the answer as an integer or a radical in simplest form. Assume that all variables represent nonnegative numbers.. 8. 9. 80. 9 81. 9 1 8. 8 81 8. p 8. z 8. r 8. 8 a 8. 8 1 88. 100 Mixed Exercises Evaluate the expression. 89. 1/ 90. / 1 91. 8 81 9. ( / ) 9. 8 1 / 9. / 9. 1/10 10 9. / 9. 1 8 1/

Chapter Eight Additional Exercises 1 98. / 99. 9 00. / 01. 1/ 100,000 0. 9 0. 1/ 0. / 100 0. / 9 / 9 9 0. / Simplify the expression. Write the answer in exponential form with only positive exponents. Assume that all variables represent positive numbers. y 0. 1/ / y y / x x 08. / x / 09. x y / / 10. / 8 1 1/ 1 11. ( / / ) 1/ r s 1. a b 1/ 1/ 1/ Writing/Conceptual Exercises Decide which one of the four choices is not equal to the given expression. 1. 1/ 11 (a). 11 (b) 11 (c) 11 (d) 11 1. / (a) (b) (c) 10 (d) 1 1. 1/ 1000 (a) 10 (b) 1 10 (c) 1 1000 (d) 0.1 1. / 8 (a) 1 (b) 8 (c) 1 1 (d)