Simplifying Radical Expressions

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1 9.2 Simplifying Radical Expressions 9.2 OBJECTIVES. Simplify expressions involving numeric radicals 2. Simplify expressions involving algebraic radicals In Section 9., we introduced the radical notation. For most applications, we will want to make sure that all radical expressions are in simplest form. To accomplish this, the following three conditions must be satisfied. Rules and Properties: Square Root Expressions in Simplest Form An expression involving square roots is in simplest form if. There are no perfect-square factors in a radical. 2. No fraction appears inside a radical.. No radical appears in the denominator. For instance, considering condition, 7 is in simplest form because 7 has no perfect-square factors whereas 2 is not in simplest form because it does contain a perfect-square factor. 2 4 To simplify radical expressions, we ll need to develop two important properties. First, look at the following expressions: Because this tells us that , the following general rule for radicals is suggested. Rules and Properties: Property of Radicals For any positive real numbers a and b, ab a b In words, the square root of a product is the product of the square roots. 707

2 708 CHAPTER 9 EXPONENTS AND RADICALS Let s see how this property is applied in simplifying expressions when radicals are involved. Example Simplifying Radical Expressions NOTE Perfect-square factors are, 4, 9, 6, 2, 6, 49, 64, 8, 00, and so on. Simplify each expression. (a) 2 4 NOTE Apply Property. NOTE Notice that we have removed the perfect-square factor from inside the radical, so the expression is in simplest form. NOTE It would not have helped to write 4 because neither factor is a perfect square. NOTE We look for the largest perfect-square factor, here 6. NOTE Then apply Property. (b) (c) (d) CAUTION Be Careful! Even though a b a b a b is not the same as a b Let a 4 and b 9, and substitute. a b 4 9 a b Because, we see that the expressions a b and a b are not in general the same. CHECK YOURSELF Simplify. (a) 20 (b) 7 (c) 98 (d) 48

3 SIMPLIFYING RADICAL EXPRESSIONS SECTION The process is the same if variables are involved in a radical expression. In our remaining work with radicals, we will assume that all variables represent positive real numbers. Example 2 Simplifying Radical Expressions Simplify each of the following radicals. (a) 2x 2x 2 x NOTE By our first rule for radicals. NOTE 2x 2 x (as long as x is positive). (b) 2x 2 x xx 24b 24 b 2 b 24b 2 b 2bb Perfect squares NOTE Notice that we want the perfect-square factor to have the largest possible even exponent, here 4. Keep in mind that a 2 a 2 a 4 (c) 28a 29 a 4 2a Perfect squares 29a 4 2a a 2 2a CHECK YOURSELF 2 Simplify. (a) 29x (b) 227m (c) 20b To develop a second property for radicals, look at the following expressions: A Because a second general rule for radicals is suggested. A 4 4,

4 70 CHAPTER 9 EXPONENTS AND RADICALS Rules and Properties: Property 2 of Radicals For any positive real numbers a and b, a a A b b In words, the square root of a quotient is the quotient of the square roots. This property is used in a fashion similar to Property in simplifying radical expressions. Remember that our second condition for a radical expression to be in simplest form states that no fraction should appear inside a radical. Example illustrates how expressions that violate that condition are simplified. Example Simplifying Radical Expressions Write each expression in simplest form. NOTE Apply Property 2 to write the numerator and denominator as separate radicals. (a) 9 A Remove any perfect squares from the radical. NOTE Apply Property 2. (b) 2 A NOTE Apply Property 2. NOTE Factor 8x 2 as 4x 2 2. NOTE Apply Property in the numerator. (c) 8x 2 B 9 28x2 9 24x2 2 24x2 2 2x2 CHECK YOURSELF Simplify x 2 (a) (b) (c) A 6 A 9 B 49

5 SIMPLIFYING RADICAL EXPRESSIONS SECTION In our previous examples, the denominator of the fraction appearing in the radical was a perfect square, and we were able to write each expression in simplest radical form by removing that perfect square from the denominator. If the denominator of the fraction in the radical is not a perfect square, we can still apply Property 2 of radicals. As we will see in Example 4, the third condition for a radical to be in simplest form is then violated, and a new technique is necessary. Example 4 Simplifying Radical Expressions Write each expression in simplest form. NOTE We begin by applying Property 2. (a) A Do you see that is still not in simplest form because of the radical in the denominator? To solve this problem, we multiply the numerator and denominator by. Note that the denominator will become 9 We then have NOTE We can do this because we are multiplying the fraction by or, which does not change its value. The expression is now in simplest form because all three of our conditions are satisfied. (b) 2 A 2 NOTE and the expression is in simplest form because again our three conditions are satisfied. NOTE We multiply numerator and denominator by 7 to clear the denominator of the radical. This is also known as rationalizing the denominator. (c) x A 7 x 7 x x 7 The expression is in simplest form.

6 72 CHAPTER 9 EXPONENTS AND RADICALS CHECK YOURSELF 4 Simplify. 2 2y (a) (b) (c) A 2 A A Both of the properties of radicals given in this section are true for cube roots, fourth roots, and so on. Here we have limited ourselves to simplifying expressions involving square roots. CHECK YOURSELF ANSWERS. (a) 2; (b) ; (c) 72; (d) 4 2. (a) xx; (b) mm; 7 2x 2 6 (c) b 2 2b. (a) ; (b) ; (c) 4. (a) ; (b) ; (c) 0y 4 7 2

7 Name 9.2 Exercises Section Date Use Property to simplify each of the following radical expressions. Assume that all variables represent positive real numbers. ANSWERS x a y x r 20. 2a b m x x

8 ANSWERS a x y y 6 2a 2 b Use Property 2 to simplify each of the following radical expressions A 2 64 A A 6 49 A A 4 A A 6 0 A Use the properties for radicals to simplify each of the following expressions. Assume that all variables represent positive real numbers a 2 7. B A A 2 2y 2 B 49 A 7 A a A 2x B 2x A 7 m 2 B 2 8s 2x B 7 B 74

9 ANSWERS Decide whether each of the following is already written in simplest form. If it is not, explain what needs to be done mn 0. 8ab x 2 y. 2. B 7x 6xy x.. Find the area and perimeter of this square: 4. One of these measures, the area, is a rational number, and the other, the perimeter, is an irrational number. Explain how this happened. Will the area always be a rational number? Explain. n 2 4. (a) Evaluate the three expressions, n, n2 using odd values of n: 2 2,,, 7, etc. Make a chart like the one below and complete it. c n2 a n2 n 2 b n 2 a 2 b 2 c (b) Check for each of these sets of three numbers to see if this statement is true: 2a 2 b 2 2c 2. For how many of your sets of three did this work? Sets of three numbers for which this statement is true are called Pythagorean triples because a 2 b 2 c 2. Can the radical equation be written in this way: 2a 2 b 2 a b? Explain your answer. 7

10 ANSWERS a. b. c. d. e. f. g. h. Getting Ready for Section 9. [Section.6] Use the distributive property to combine the like terms in each of the following expressions. (a) x 6x (c) 0y 2y (e) 9a 7a 2a (g) 2m n 6m Answers (b) 8a a (d) 7m 0m (f) s 8s 4s (h) 8x y 4x x 7. y 2 9. r2r 2. b x a 2 6a 27. xyx a2 6 a x s4s Simplest form 7. Remove the perfect-square factors from the radical and simplify.. a. x b. a c. 2y d. 7m e. 4a f. s g. 6m n h. 4x y 76

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