9.3 OPERATIONS WITH RADICALS


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1 9. Operations with Radicals (9 1) OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in performing arithmetic operations with radical epressions. Adding and Subtracting Radicals To find the sum of and, we can use a calculator to get 1.1 and 1.7. (The symbol means is approimately equal to. ) We can then add the decimal numbers and get We cannot write an eact decimal form for ; the number.16 is an approimation of. To represent the eact value of, we just use the form. This form cannot be simplified any further. However, a sum of like radicals can be simplified. Like radicals are radicals that have the same inde and the same radicand. To simplify the sum, we can use the fact that 8 is true for any value of. Substituting for gives us 8. So like radicals can be combined just as like terms are combined. E X A M P L E 1 Adding and subtracting like radicals Simplify the following epressions. Assume the variables represent positive numbers. a) b) w 6 w c) 6 d) 6 6 a) 7 b) w 6 w w c) 6 7 d) Only like radicals are combined. Remember that only radicals with the same inde and same radicand can be combined by addition or subtraction. If the radicals are not in simplified form, then they must be simplified before you can determine whether they can be combined. E X A M P L E Simplifying radicals before combining Perform the indicated operations. Assume the variables represent positive numbers. a) 8 18 b) 1 0 c) 18 d) 16 y y a) Simplify each radical. Add like radicals. Note that
2 88 (9 ) Chapter 9 Radicals and Rational Eponents calculator closeup Check that =. b) 1 0 Because 1 1 and 0 10 Use the LCD of. 11 c) d) 16 y y 8 y 7 y y y y Simplify each radical. Simplify each radical. Add like radicals only. Multiplying Radicals We have already multiplied radicals in Section 9., when we rationalized denominators. The product rule for radicals, n a n b n ab, allows multiplication of radicals with the same inde, such as 1, 10, and. CAUTION The product rule does not allow multiplication of radicals that have different indices. We cannot use the product rule to multiply and. E X A M P L E helpful hint Students often write Although this is correct, you should get used to the idea that Because of the definition of a square root, a a a for any positive number a. Multiplying radicals with the same inde Multiply and simplify the following epressions. Assume the variables represent positive numbers. a) 6 b) a 6a c) d) a) Product rule for radicals b) a 6a 18a Product rule for radicals c) 16 9a a aa Simplify. 8 Simplify.
3 9. Operations with Radicals (9 ) 89 d) Product rule for radicals Simplify Rationalize the denominator We find a product such as ( ) by using the distributive property as we do when multiplying a monomial and a binomial. A product such as ( )( ) can be found by using FOIL as we do for the product of two binomials. E X A M P L E Multiplying radicals Multiply and simplify. a) ( ) b) a ( a a) c) ( )( ) d) ( ) 9 a) ( ) Distributive property 1 6 Because and 6 6 b) a ( a a) a a a a c) ( )( ) Distributive property F O I L Combine like radicals. d) To square a sum, we use (a b) a ab b : ( 9 ) 9 ) ( In the net eample we multiply radicals that have different indices.
4 90 (9 ) Chapter 9 Radicals and Rational Eponents E X A M P L E calculator closeup Check that Multiplying radicals with different indices Write each product as a single radical epression. a) b) a) 1 1 Write in eponential notation. 71 Product rule for eponents: Write in radical notation b) 1 1 Write in eponential notation. 6 6 Write the eponents with the LCD of Write in radical notation. 6 Product rule for radicals CAUTION Because the bases in 1 1 are identical, we can add the eponents [Eample (a)]. Because the bases in 6 6 are not the same, we cannot add the eponents [Eample (b)]. Instead, we write each factor as a sith root and use the product rule for radicals. helpful hint The word conjugate is used in many contets in mathematics. According to the dictionary, conjugate means joined together, especially as in a pair. Conjugates Recall the special product rule (a b)(a b) a b. The product of the sum and the difference can be found by using this rule: ( )( ) () 16 1 The product of the irrational number and the irrational number is the rational number 1. For this reason the epressions and are called conjugates of one another. We will use conjugates in Section 9. to rationalize some denominators. E X A M P L E 6 Multiplying conjugates Find the products. Assume the variables represent positive real numbers. a) ( )( ) b) ( )( ) c) ( y)( y) a) ( )( ) () (a b)(a b) a b () 9 1 b) ( )( ) 1 c) ( y)( y) y
5 9. Operations with Radicals (9 ) 91 WARMUPS True or false? Eplain your answer ( ) ( ) 10. ( )( ) 1 9. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What are like radicals?. How do we combine like radicals?. Does the product rule allow multiplication of unlike radicals?. How do we multiply radicals of different indices? All variables in the following eercises represent positive numbers. Simplify the sums and differences. Give eact answers. See Eample a 76a y y ab a a ab Simplify each epression. Give eact answers. See Eample
6 9 (9 6) Chapter 9 Radicals and Rational Eponents. t y 16t y w z 16w z Simplify the products. Give eact answers. See Eamples and ()(10) 1. 7a a. c () 6. () (6 ) 0. ( ) 1. (10 ). 6(1 1). t ( 9t t). ( 1 ). ( )( ) 6. ( )( 6) 7. (11 )(11 ) 8. ( )( ) 9. ( 7)( ) 60. (6 )(6 ) 61. ( 6)( 6) 6. ( )( ) Write each product as a single radical epression. See Eample Find the product of each pair of conjugates. See Eample ( )( ) 7. (7 )(7 ) 7. ( )( ) 7. (6 )(6 ) 7. ( 1)( 1) 76. ( )( ) 77. ( )( ) 78. ( 7)( 7) 79. ( )( ) 80. (y z)(y z) Simplify each epression ( 7)(7 ) 86. ( 7)(7 ) 87. w w 88. m m t 10t ( )( ) 9. ( )( ) ( )( ) 98. ( 7)( 7) 99. ( ) 100. (1 ) 101. ( ) 10. (a ) 10. (1 ) 10. ( 1 1) 10. w 9w m 16m 107. a a aa 108. w y 7w y 6w y y 7 y 7 7 y z 11.
7 9. More Operations with Radicals (9 7) a( a a) m n In Eercises 119 1, solve each problem Area of a rectangle. Find the eact area of a rectangle that has a length of 6 feet and a width of feet. 10. Volume of a cube. Find the eact volume of a cube with sides of length meters. 1. Area of a triangle. Find the eact area of a triangle with a base of 0 meters and a height of 6 meters. 6 m 0 m FIGURE FOR EXERCISE 1 m m FIGURE FOR EXERCISE Area of a trapezoid. Find the eact area of a trapezoid with a height of 6 feet and bases of feet and 1 feet. ft 6 ft 1 ft m FIGURE FOR EXERCISE 11 GETTING MORE INVOLVED 1. Discussion. Is a b a b for all values of a and b? 1. Discussion. Which of the following equations are identities? Eplain your answers. a) 9 b) 9 c) d) 1. Eploration. Because is the square of, a binomial such as y is a difference of two squares. a) Factor y and a 7 using radicals. b) Use factoring with radicals to solve the equations 8 0 and y c) Assuming a and b are positive real numbers, solve the equations a 0 and a b MORE OPERATIONS WITH RADICALS In this section In this section you will continue studying operations with radicals. We learn to rationalize some denominators that are different from those rationalized in Section 9.. Dividing Radicals Rationalizing the Denominator Powers of Radical Epressions Dividing Radicals In Section 9. you learned how to add, subtract, and multiply radical epressions. To divide two radical epressions, simply write the quotient as a ratio and then simplify, as we did in Section 9.. In general, we have n a n b n a n b n a b, provided that all epressions represent real numbers. Note that the quotient rule is applied only to radicals that have the same inde.
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