Simplification of Radical Expressions


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1 8. Simplification of Radical Expressions 8. OBJECTIVES 1. Simplify a radical expression by using the product property. Simplify a radical expression by using the quotient property NOTE A precise set of conditions for a radical to be in simplified form will follow in this section. In the last section, we introduced the radical notation. For some applications, we will want to make sure that all radical expressions are written in simplified form. To accomplish this objective, we will need two basic properties. In stating these properties, and in our subsequent examples, we will assume that all variables represent positive real numbers whenever the index of a radical is even. To develop our first property, consider an expression such as 1 4 One approach to simplify the expression would be Now what happens if we separate the original radical as follows? The result in either case is the same, and this suggests our first property for radicals. Rules and Properties: Product Property for Radicals NOTE As we stated in the first paragraph, a and b are assumed to be positive real numbers when n is an even integer. 1 n ab 1 n a 1 n b In words, the radical of a product is equal to the product of the radicals. The second property we will need is similar. Rules and Properties: Quotient Property for Radicals NOTE To convince yourself that this must be the case, at least for square roots, let a 100 and b 4 and evaluate both sides of the equation. CAUTION NOTE You can easily see that this is not true. Let a 9 and b 16 in the statement. n a Ab 1n a 1 n b In words, the radical of a quotient is the quotient of the radicals. Be Careful! Students sometimes assume that because 1ab 1a 1b it should also be true that 1a b 1a 1b This is not true. 91
2 9 CHAPTER 8 RADICAL EXPRESSIONS With these two properties, we are now ready to define the simplified form for a radical expression. A radical is in simplified form if the following three conditions are satisfied. Definitions: Simplified Form for a Radical Expression 1. The radicand has no factor raised to a power greater than or equal to the index.. No fraction appears in the radical.. No radical appears in a denominator. Our initial example deals with satisfying the first of the above conditions. Essentially, we want to find the largest perfectsquare factor (in the case of a square root) in the radicand and then apply the product property to simplify the expression. Example 1 Simplifying Radical Expressions NOTE The largest perfectsquare factor of 18 is Apply the product property. 1 NOTE The largest perfectsquare factor of is NOTE The largest perfectsquare factor of x is 9x. Note that the exponent must be even in a perfect square. (c) (d) x 9x x 9x 1x x1x a b 4 6a b 4 a 6a b 4 1a 6ab 1a CHECK YOURSELF (c) p (d) 98m n 4 Writing a cube root in simplest form involves finding factors of the radicand that are perfect cubes, as illustrated in Example. The process illustrated in this example is extended in an identical fashion to simplify radical expressions with any index.
3 SIMPLIFICATION OF RADICAL EXPRESSIONS SECTION 8. 9 Example Simplifying Radical Expressions NOTE In a perfect cube, the exponent must be a multiple of. 4x 4 8x x 8x 1 x x1 x (c) 4a b 4 a 6 b ab a 6 b 1 ab a b1 ab CHECK YOURSELF 18w 4 40x y (c) 4 48a 8 b Satisfying our second condition for a radical to be in simplified form (no fractions should appear inside the radical) requires the second property for radicals. Consider the following example. Example Simplifying Radical Expressions NOTE Apply the quotient property. A 9 a 4 B a4 1 a (c) x 1 8 x x B 8 CHECK YOURSELF x (c) A 16 A a A
4 94 CHAPTER 8 RADICAL EXPRESSIONS Our next example also begins with the application of the quotient property for radicals. However, an additional step is required because, as we will see, the third condition (no radicals can appear in a denominator) must also be satisfied during the process. Example 4 Rationalizing the Denominator Write B A 1 1 in simplified form. NOTE The value of the expression is not changed as we 1 multiply by or 1. 1, NOTE The point here is to arrive at a perfect square inside the radical in the denominator. This is done by multiplying the numerator and denominator by 1 because The application of the quotient property satisfies the second condition there are now no fractions inside a radical. However, we now have a radical in the denominator, violating the third condition. The expression will not be simplified until that radical is removed. To remove the radical in the denominator, we multiply the numerator and denominator by the same expression, here 1. This is called rationalizing the denominator CHECK YOURSELF 4 Simplify A Let s look at some further examples that involve rationalizing the denominator of an expression. NOTE We multiply numerator and denominator by 1. Why did we choose 1? Note that 18 so Example Rationalizing the Denominator
5 SIMPLIFICATION OF RADICAL EXPRESSIONS SECTION 8. 9 A Now note that NOTE Why did we use Note that ? and the exponent is a multiple of so multiplying the numerator and denominator by 1 will produce a perfect cube inside the radical in the denominator. Continuing, we have CHECK YOURSELF Simplify each expression. 11 A 9 As our final example, we illustrate the process of rationalizing a denominator when variables are involved in a rational expression. Example 6 Rationalizing Variable Denominators Simplify each expression. 8x B y By the quotient property we have 8x B y 8x 1y Because the numerator can be simplified in this case, let s start with that procedure. 8x 1y 4x 1x x1x 1y 1y Multiplying the numerator and denominator by 1y will rationalize the denominator. x1x 1y 1y 1y x16xy 9y x16xy y 1 x To satisfy the third condition, we must remove the radical from the denominator. For this we need a perfect cube inside the radical in the denominator. Multiplying the numerator
6 96 CHAPTER 8 RADICAL EXPRESSIONS NOTE 9x x so 1 x 9x x and each exponent is a multiple of. and denominator by 9x will provide the perfect cube. So 9x 9x 1 x 9x x 9x x CHECK YOURSELF 6 Simplify each expression. 1a B b w The following algorithm summarizes our work in simplifying radical expressions. Step by Step: Simplifying Radical Expressions NOTE In the case of a cube root, steps 1 and would refer to perfect cubes, etc. Step 1 Step Step To satisfy the first condition: Determine the largest perfectsquare factor of the radicand. Apply the product property to remove that factor from inside the radical. To satisfy the second condition: Use the quotient property to write the expression in the form 1a 1b If b is a perfect square, remove the radical in the denominator. If that is not the case, proceed to step. Multiply the numerator and denominator of the radical expression by an appropriate radical to remove the radical in the denominator. Simplify the resulting expression when necessary. CHECK YOURSELF ANSWERS 1. 1; 101; (c) p 1p; (d) mn 1m w1 w; xy x y; (c) a b1 4 b. (c) 4 ; a ; a11ab w ; 6 ; b w 1 x
7 Name 8. Exercises Section Date Use the product property to write each expression in simplified form ANSWERS Use the product property to write each expression in simplified form. Assume that all variables represent positive real numbers z 8. 4a 9. 6x w m. a. 80x y p q. 40b 6. 16x. 48p a
8 ANSWERS m 40. 0x r 6 s 4. 6x 6 y z a b 4 0a 4 b 1 c x y a p w z a 1 b w a b Use the quotient property to write each expression in simplified form. Assume that all variables represent positive real numbers A 16 A 6 x 4 B B 49 A 9y 4 a 6 A x A 8 A 64 B 4x B 4 x 16 A 4 81a 8 A 8y Assume that all variables represent positive real numbers A A A
9 ANSWERS A 4 A A x A y A w 118 1a m 4x n B y A y A x 1 x 1 a A x A w 1 4a B a 9m b B w z Label each of the following statements as true or false x 16 4x x y x y x 1x 1x (8b 6 ) 8b x 6 x 1 x 1 x 1 8x 1 x 4x
10 ANSWERS Simplify. x y 4 16xy x 6 y 49x 1 y c 1 d c d 4 4 9c 8 d c d (c) (d) 10. Explain the difference between a pair of binomials in which the middle sign is changed, and the opposite of a binomial. To illustrate, use Determine the missing binomial in the following: (1 )( ) Use a calculator to evaluate the following expressions in parts through (d). Round your answer to the nearest hundredth (c) (d) 16 (e) Based on parts through (d) make a conjecture concerning a1m b1m. Check your conjecture on an example of your own similar to parts through (d). Answers z1 9. x 1 1. m1m. 4xy1y. b1. p m 1 m 41. ab a b 4. x yz y z 4. x a 4 a 1 x wz 4 6wz 1. w y 4x a x x m110mn y 4x 1 0x n y x x a a b True 9. True 99. True 101. x y b ; 1.6; (c) 1.; (d) 1. 1w w 1 10a a
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