7.5 Rationalizing Denominators and Numerators of Radical Expressions

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1 440 CHAPTER 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 meters.) m 6 m m 6 m 8. Add: +. b. Multiply: #. c. Describe the differences in parts (a) and (b). 88. 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 x - x + is similar to multiplying x - x +.. Rationalizing Denominators and Numerators of Radical Expressions S Rationalize Denominators. Rationalize Denominators Having Two Terms. Rationalize Numerators. Rationalizing Denominators of Radical Expressions Often in mathematics, it is helpful to write a radical expression such as 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, we use the fundamental principle of fractions and multiply the numerator and the denominator by. Recall that this is the same as multiplying by, which simplifies to. # # In this section, we continue to assume that variables represent positive real numbers. EXAMPLE Solution Rationalize the denominator of each expression. b. 6 9x c. 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. 6 9x 4 x 8 x To rationalize the denominator, multiply the numerator and denominator by x. Then 8 x 8 # x x # x 8x x

2 Section. Rationalizing Denominators and Numerators of Radical Expressions 44 c. A. Now we rationalize the denominator. Since is a cube root, we want to multiply by a value that will make the radicand a perfect cube. If we multiply by, we get # # Rationalize the denominator of each expression. b. 4x Multiply the numerator and denominator by and then simplify. c. A 9 CONCEPT CHECK Determine by which number both the numerator and denominator can be multiplied to rationalize the denominator of the radical expression. b. 4 8 EXAMPLE Solution x A y x y Rationalize the denominator of A x y. x # y y # y xy y Rationalize the denominator of A z y. Use the quotient rule. No radical may be simplified further. Multiply numerator and denominator by y so that the radicand in the denominator is a perfect square. Use the product rule in the numerator and denominator. Remember that y # y y. EXAMPLE Rationalize the denominator of Solution First, simplify each radical if possible. 4 x 4 x 4 8y 4 8y 4 # 4 y 4 x y 4 y 4 x # 4 y y 4 y # 4 y 4 xy y 4 y 4 4 xy y 4 x 4 8y. Use the product rule in the denominator. Write 4 8y 4 as y. Multiply numerator and denominator by 4 y so that the radicand in the denominator is a perfect fourth power. Use the product rule in the numerator and denominator. In the denominator, 4 y 4 y and y # y y. Answer to Concept Check: or 49 b. 4 Rationalize the denominator of z x 4.

3 44 CHAPTER Rational Exponents, Radicals, and Complex Numbers Rationalizing Denominators Having Two Terms Remember the product of the sum and difference of two terms? a + ba - b a - b a Q These two expressions are called conjugates of each other. To rationalize a numerator or denominator that is a sum or difference of two terms, we use conjugates. To see how and why this works, let s rationalize the denominator of the expression. To do so, we multiply both the numerator and the denominator - by +, the conjugate of the denominator -, and see what happens Multiply the sum and difference of two terms: a + ba - b a - b. - + or Notice in the denominator that the product of - and its conjugate, +, is -. In general, the product of an expression and its conjugate will contain no radical terms. This is why, when rationalizing a denominator or a numerator containing two terms, we multiply by its conjugate. Examples of conjugates are a - b and a + b x + y and x - y EXAMPLE Solution Rationalize each denominator. b c. m x + m Multiply the numerator and denominator by the conjugate of the denominator, , or - 4 It is often useful to leave a numerator in factored form to help determine whether the expression can be simplified.

4 Section. Rationalizing Denominators and Numerators of Radical Expressions 44 b. Multiply the numerator and denominator by the conjugate of c. Multiply by the conjugate of x + m to eliminate the radicals from the denominator. m x + m mx - m x + mx - m 6mx - m 9x - m 6mx - m x - m 4 Rationalize the denominator. + b. + - c. x x + y Rationalizing Numerators As mentioned earlier, it is also often helpful to write an expression such as as an equivalent expression without a radical in the numerator. This process is called rationalizing the numerator. To rationalize the numerator of, we multiply the numerator and the denominator by. # # EXAMPLE Rationalize the numerator of 4. Solution First we simplify # Next we rationalize the numerator by multiplying the numerator and the denominator by. # # # Rationalize the numerator of 80.

5 444 CHAPTER Rational Exponents, Radicals, and Complex Numbers EXAMPLE 6 Rationalize the numerator of x y. Solution The numerator and the denominator of this expression are already simplified. To rationalize the numerator, x, we multiply the numerator and denominator by a factor that will make the radicand a perfect cube. If we multiply x by 4x, we get 8x x. x y x # 4x y # 4x 8x 0xy x 0xy 6 Rationalize the numerator of b a. EXAMPLE Rationalize the numerator of x +. Solution We multiply the numerator and the denominator by the conjugate of the numerator, x +. x + x + x - x - x - x - x - 4 x - Rationalize the numerator of x - 4 Multiply by x -, the conjugate of x +. a + ba - b a - b. Vocabulary, Readiness & Video Check Use the choices below to fill in each blank. Not all choices will be used. rationalizing the numerator conjugate rationalizing the denominator. The of a + b is a - b.. The process of writing an equivalent expression, but without a radical in the denominator is, called.. The process of writing an equivalent expression, but without a radical in the numerator, is called. 4. To rationalize the denominator of, we multiply by. Martin-Gay Interactive Videos See Video. Watch the section lecture video and answer the following questions.. From Examples, what is the goal of rationalizing a denominator? 6. From Example 4, why will multiplying a denominator by its conjugate always rationalize the denominator?. From Example, is the process of rationalizing a numerator any different from rationalizing a denominator?

6 Section. Rationalizing Denominators and Numerators of Radical Expressions 44. Exercise Set Rationalize each denominator. See Examples through A 4 A x 4 8x x. 9 a x A y A x A 0 z y 4. A A y 6 9 a y x x A A9x 0. - a A b A0 y B 4 x x 4y 4 4 A9. Am 6 n a 9y. 4. 8a 9 b 4 4y 9 Write the conjugate of each expression.. + x 6. + y. - a b x y Rationalize each denominator. See Example x y a + a - b x x + y a - a + b a x - y Rationalize each numerator. See Examples and A 8 A 4x y 4x A x A8 x 0 8x 4 y 6 B z A A x 6 4x z 4 A y A 9y A 8x y B z Rationalize each numerator. See Example x + x - + x + x x x + y x - y

7 446 CHAPTER Rational Exponents, Radicals, and Complex Numbers REVIEW AND PREVIEW Solve each equation. See Sections. and x - x x - 4 x - 8. x - 6x y + y x - 8x x x CONCEPT EXTENSIONS 89. The formula of the radius r of a sphere with surface area A is A r A 4p Rationalize the denominator of the radical expression in this formul 90. The formula for the radius r of a cone with height centimeters and volume V is V r A p Rationalize the numerator of the radical expression in this formul r cm r y 9. Given rationalize the denominator by following x, parts (a) and (b). Multiply the numerator and denominator by x. b. Multiply the numerator and denominator by x. c. What can you conclude from parts (a) and (b)? 9. Given y, rationalize the denominator by following parts 4 (a) and (b). Multiply the numerator and denominator by 6. b. Multiply the numerator and denominator by. c. What can you conclude from parts (a) and (b)? Determine the smallest number both the numerator and denominator should be multiplied by to rationalize the denominator of the radical expression. See the Concept Check in this section When rationalizing the denominator of, explain why both the numerator and the denominator must be multiplied by. 96. When rationalizing the numerator of, explain why both the numerator and the denominator must be multiplied by. 9. Explain why rationalizing the denominator does not change the value of the original expression. 98. Explain why rationalizing the numerator does not change the value of the original expression. Integrated Review RADICALS AND RATIONAL EXPONENTS Sections.. Throughout this review, assume that all variables represent positive real numbers. Find each root A x 6. y y 0. -y b Use radical notation to write each expression. Simplify if possible / 0. y / /. x + / Use the properties of exponents to simplify each expression. Write with positive exponents.. y -/6 # y /6 4. x/ 4 x /6. x/4 x /4 x -/ / # 4 / Use rational exponents to simplify each radical.. 8x 6 8. a 9 b 6

8 Section.6 Radical Equations and Problem Solving 44 Use rational exponents to write each as a single radical expression x # x 0. # Simplify x y 0. 4x b 0 Multiply or divide. Then simplify if possible.. # x 6. 8x # 8x. 98y a 9 b y 4 ab Perform each indicated operation y 4 - y 6y x - x + 4. x + - Rationalize each denominator.. A Rationalize each numerator. 8. A 6. x. 9. A 9y x - x.6 Radical Equations and Problem Solving S Solve Equations That Contain Radical Expressions. Use the Pythagorean Theorem to Model Problems. Solving Equations That Contain Radical Expressions In this section, we present techniques to solve equations containing radical expressions such as x - 9 We use the power rule to help us solve these radical equations. Power Rule If both sides of an equation are raised to the same power, all solutions of the original equation are among the solutions of the new equation. This property does not say that raising both sides of an equation to a power yields an equivalent equation. A solution of the new equation may or may not be a solution of the original equation. For example, -, but -. Thus, each solution of the new equation must be checked to make sure it is a solution of the original equation. Recall that a proposed solution that is not a solution of the original equation is called an extraneous solution. EXAMPLE Solve: x - 9. Solution We use the power rule to square both sides of the equation to eliminate the radical. x - 9 x - 9 x - 8 x 84 x 4 Now we check the solution in the original equation. (Continued on next page)

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