7.4 Adding, Subtracting, and Multiplying Radical Expressions. OBJECTIVES 1 Add or Subtract Radical Expressions. 2 Multiply Radical Expressions.

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1 CHAPTER 7 Rational Exponents, Radicals, and Complex Numbers Find and correct the error. See the Concept Check in this section = 6 A6 = 1 = 1 16 = 16 A = Simplify. See a Concept Check in this section. Assume variables represent positive numbers x y a 1 b c a 9 b 1 c 11. z 1. x q 17 r 0 s 7 1. p 11 q r 1. The formula for the radius r of a sphere with surface area A A is given by r =. Calculate the radius of a standard A p zorb whose outside surface area is.17 sq m. Round to the nearest tenth. (A zorb is a large inflated ball within a ball in which a person, strapped inside, may choose to roll down a hill. Source: Zorb, Ltd.) a. Approximate to one decimal place the demand per week of an older released DVD if the rental price is $ per two-day rental. b. Approximate to one decimal place the demand per week of an older released DVD if the rental price is $ per two-day rental. c. Explain how the owner of the store can use this equation to predict the number of copies of each DVD that should be in stock. 17. The formula for the lateral surface area A of a cone with height h and radius r is given by A = prr + h a. Find the lateral surface area of a cone whose height is centimeters and whose radius is centimeters. b. Approximate to two decimal places the lateral surface area of a cone whose height is 7. feet and whose radius is 6.8 feet. h r 18. Before Mount Vesuvius, a volcano in Italy, erupted violently in 79 c.e., its height was 190 feet. Vesuvius was roughly cone-shaped, and its base had a radius of approximately,00 feet. Use the formula for the lateral surface area of a cone, given in Exercise 17, to approximate the surface area this volcano had before it erupted. (Source: Global Volcanism Network) 16. The owner of Knightime Classic Movie Rentals has determined that the demand equation for renting older released DVDs is F1x = x, where x is the price in dollars per two-day rental and F1x is the number of times the DVD is demanded per week. 190 ft,00 ft 7. Adding, Subtracting, and Multiplying Radical Expressions S 1 Add or Subtract Radical Expressions. Multiply Radical Expressions. 1 Adding or Subtracting Radical Expressions We have learned that sums or differences of like terms can be simplified. To simplify these sums or differences, we use the distributive property. For example, x + x = 1 + x = x and 7x y - x y = 17 - x y = x y The distributive property can also be used to add like radicals. Like Radicals Radicals with the same index and the same radicand are like radicals.

2 Section 7. Adding, Subtracting, and Multiplying Radical Expressions For example, = = 7. Also, Like radicals x - 7x = 1-7 x = -x The expression cannot be simplified further since 7 and 7are not like radicals. Unlike radicals EXAMPLE 1 Add or subtract as indicated. Assume all variables represent positive real numbers. a b. x - 7 x c. + a = = 111 b. x - 7 x = 1-7 x = - x c. + This expression cannot be simplified since and do not contain like radicals. 1 Add or subtract as indicated. a b. 7 z - 1 z c. + When adding or subtracting radicals, always check first to see whether any radicals can be simplified. CONCEPT CHECK True or false? Explain. a + b = a + b Answer to Concept Check: false; answers may vary EXAMPLE Add or subtract. Assume that variables represent positive real numbers. a. 0 + b c. 7x - 9x + 7x d e. 8y + 6y First, simplify each radical. Then add or subtract any like radicals. a. 0 + = # + 9 # Factor 0 and. = # + # 9 # Use the product rule. = # + # # Simplify and 9. = + 6 = 8 Add like radicals. b = 7 # - # 8 # + Factor and use the product rule. = # - # # + Simplify 7 and 8. = Write # as 10. = -6 Combine like radicals. (Continued on next page)

3 6 CHAPTER 7 Rational Exponents, Radicals, and Complex Numbers Helpful Hint None of these terms contain like radicals. We can simplify no further. c. 7x - 9x + 7x = 9 # x - # 9 # x + 6 # x Factor and use the product rule. = # x - # # x + 6 # x Simplify 9 and 6. = x - 6x + 6x Write # as 6. d = # Factor and use the product rule. = No further simplification is possible. e. 8y + 6y = 8y # 6y + y # 6y Factor and use the product rule. = y 6y + y 6y Simplify 8y and y. = y 6y Combine like radicals. Add or subtract. a. + b c. 7x - 7x + 1x d e. 81x + x Let s continue to assume that variables represent positive real numbers. EXAMPLE a. a. - - = Add or subtract as indicated. - b. A 7x 8 + 7x To subtract, notice that the LCD is 1. = # # - # Write each expression as an equivalent # expression with a denominator of 1. = 9 1 = 1-1 Multiply factors in the numerator and the denominator. Subtract. b. A 7x 8 + 7x = 7x 8 + 7x Apply the quotient rule for radicals. = 7x = 7x = 7x = 7x + 7x Simplify. + 7x # + 7x Add or subtract as indicated. a. 8-7 Write each expression as an equivalent expression with a denominator of. Add. b. A 6y 6 + 6y

4 Section 7. Adding, Subtracting, and Multiplying Radical Expressions 7 Multiplying Radical Expressions We can multiply radical expressions by using many of the same properties used to multiply polynomial expressions. For instance, to multiply , we use the distributive property and multiply 1 by each term inside the parentheses = Use the distributive property. EXAMPLE Multiply. a b c. 17x + 1x - d. 1-1 e. 1 x - 1x + f. 1 x - + a = = + # 0 = + # # 10 = + 10 b. To multiply, we can use the FOIL method. First Outer Inner Last = # 7 + # 1-6 # 7-6 # 1 = c. 17x + 1x - = 7x1x - 7x1 + 1x - 1 = 1x - 7x + 1x - d. 1-1 = = = 16 # = = 9-8 e. 1 x - 1x + = x # x + x - x - # = x - f. 1 x - + = 1 x - + # x - # + c a c b Multiply. = # 6 - # c a c + c c # a c c # b + b = x x - + Simplify. = x x - Combine like terms. a b c. 1z - 1z + d = # # - # Use the product rule for radicals. = - 6 e. 1 x + 1x - f. 1 x + +

5 8 CHAPTER 7 Rational Exponents, Radicals, and Complex Numbers Vocabulary, Readiness & Video Check Complete the table with Like or Unlike. Terms Like or Unlike Radical Terms? 1. 7, 7. x y, yx. abc, cba. x, x10 Simplify. Assume that all variables represent positive real numbers.. + = = 7. 8x - x = 8. y - y = 9. 7 x + x = z + z = Martin-Gay Interactive Videos Watch the section lecture video and answer the following questions From Examples 1 and, why should you always check to see if all terms in your expression are simplified before attempting to add or subtract radicals? 1. In Example, what are you told to remember about the square root of a positive number? See Video Exercise Set Add or subtract. See Examples 1 through x + x8x. x + xx x - x 8. a - a 81a 9. 9b - b + 9b 10. x 7 + 9x x - xx A x 9 x7-1 + A x 9 + B 7x y - 6y x x a 9ab - a 7 b + 16a 7 b 6. x 7 y + 9x x y - xyx y

6 Section 7. Adding, Subtracting, and Multiplying Radical Expressions 9 7. y8y + 0y 8. 8x y - xy 9. xy - xy + y 18x 0. x y + x 81y x x 7 - x x. 6 x - 81x - x x 7 8 x 8 A x 99 x x x 7-10x + 7 A x - A x 16 A A1. - x 9 + B 0x 7. y + y y 8. Find the perimeter of the trapezoid. in. 7 in. 1 in. 6. Find the perimeter of the triangle. 8 m m m 1 in. Multiply and then simplify if possible. See Example x - 1x - 1. x1 - x. y1 y +. 1x - 1x y + z1y a - 1 a a + 1 a x x x - y1x + y 6. 1 x + 1x x y - x x - x1x - x x - x17x + 6x x + 11 x - x x + 1 9x - x x x x x REVIEW AND PREVIEW Factor each numerator and denominator. Then simplify if possible. See Section x x - y 7x - 7y x - y 80. x - 8 x - 8 6a b - 9ab ab CONCEPT EXTENSIONS r - 8r s 7rs Find the perimeter and area of the rectangle. 0 ft 1 ft

7 0 CHAPTER 7 Rational Exponents, Radicals, and Complex Numbers 86. Find the area and perimeter of the trapezoid. (Hint: The area of a trapezoid is the product of half the height 6 meters and the sum of the bases 6 and 77 meters.) 7 m 6 m 7 7 m 6 m 87. a. Add: +. b. Multiply: #. c. Describe the differences in parts (a) and (b). 88. a. Add: + b. Multiply: # c. Describe the differences in parts (a) and (b). 89. Multiply: Multiply: Explain how simplifying x + x is similar to simplifying x + x. 9. Explain how multiplying 1x - 1x + is similar to multiplying 1 x - 1x Rationalizing Denominators and Numerators of Radical Expressions S 1 Rationalize Denominators. Rationalize Denominators Having Two Terms. Rationalize Numerators. 1 Rationalizing Denominators of Radical Expressions Often in mathematics, it is helpful to write a radical expression such as 1 1 either without a radical in the denominator or without a radical in the numerator. The process of writing this expression as an equivalent expression but without a radical in the denominator is called rationalizing the denominator. To rationalize the denominator of 1, we use the fundamental principle of fractions and multiply the numerator and 1 the denominator by. Recall that this is the same as multiplying by 1 1, which simplifies to 1. = # # = 6 = 6 In this section, we continue to assume that variables represent positive real numbers. a. EXAMPLE 1 Rationalize the denominator of each expression. b. 16 9x c. A 1 a. To rationalize the denominator, we multiply the numerator and denominator by a factor that makes the radicand in the denominator a perfect square. = # # = The denominator is now rationalized. b. First, we simplify the radicals and then rationalize the denominator. 16 9x = 1 x = 8 x To rationalize the denominator, multiply the numerator and denominator by x. Then 8 x = 8 # x x # x = 8x x