Multiplying and Dividing Radicals


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1 9.4 Multiplying and Dividing Radicals 9.4 OBJECTIVES 1. Multiply and divide expressions involving numeric radicals 2. Multiply and divide expressions involving algebraic radicals In Section 9.2 we stated the first property for radicals: 1ab 1a 1b when a and b are any positive real numbers That property has been used to simplify radical expressions up to this point. Suppose now that we want to find a product, such as We can use our first radical rule in the opposite manner. NOTE The product of square roots is equal to the square root of the product of the radicands. 1a 1b 1ab so We may have to simplify after multiplying, as Example 1 illustrates. Example 1 Simplifying Radical Expressions Multiply then simplify each expression. (a) (b) An alternative approach would be to simplify 112 first (c) 110x 12x 220x 2 24x x x15 72
2 724 CHAPTER 9 EXPONENTS AND RADICALS CHECK YOURSELF 1 Simplify. (a) 1 16 (b) (c) 18a 1a If coefficients are involved in a product, we can use the commutative and associative properties to change the order and grouping of the factors. This is illustrated in Example 2. Example 2 Multiplying Radical Expressions NOTE In practice, it is not necessary to show the intermediate steps. (215)(16) (2 )(15 16) CHECK YOURSELF 2 Multiply ( 17)( 51). The distributive property can also be applied in multiplying radical expressions. Consider the following. Example Multiplying Radical Expressions (a) 1(12 1) The distributive property Multiply the radicals. (b) 15(216 1) The distributive property The commutative property CHECK YOURSELF (a) 15(16 15) (b) 1(215 12) The FOIL pattern we used for multiplying binomials in Section.4 can also be applied in multiplying radical expressions. This is shown in Example 4.
3 MULTIPLYING AND DIVIDING RADICALS SECTION Example 4 Multiplying Radical Expressions (a) (1 2)(1 5) Combine like terms CAUTION NOTE You can use the pattern (a b)(a b) a 2 b 2, where a 17 and b 2, for the same result and 17 2 are called conjugates of each other. Note that their product is the rational number. The product of conjugates will always be rational. Be Careful! This result cannot be further simplified: 1 and 71 are not like terms. (b) (c) (17 2)(17 2) (1 5) 2 (1 5)(1 5) CHECK YOURSELF 4 (a) (15 )(15 2) (b) (1 4)(1 4) (c) (12 ) 2 We can also use our second property for radicals in the opposite manner. NOTE The quotient of square roots is equal to the square root of the quotient of the radicands. 1a 1b a A b One use of this property to divide radical expressions is illustrated in Example 5. Example 5 Simplifying Radical Expressions Simplify. NOTE The clue to recognizing when to use this approach is in noting that 48 is divisible by. (a) (b) A A x 2 125x 2 (c) 225x 2 5x 15 A 5 There is one final quotient form that you may encounter in simplifying expressions, and it will be extremely important in our work with quadratic equations in the next chapter. This form is shown in Example 6.
4 726 CHAPTER 9 EXPONENTS AND RADICALS CHECK YOURSELF 5 Simplify. (a) (b) 281s2 19 Example 6 Simplifying Radical Expressions Simplify the expression 172 CAUTION First, we must simplify the radical in the numerator. Be Careful! Students are sometimes tempted to write This is not correct. We must divide both terms of the numerator by the common factor (1 212) Use Property 1 to simplify 172. Factor the numerator then divide by the common factor. CHECK YOURSELF 6 Simplify CHECK YOURSELF ANSWERS 1. (a) 12; (b) 16; (c) 2a (a) 10 5; (b) (a) 1 15; (b) 1; (c) (a) 5; (b) s 6. 1
5 Name 9.4 Exercises Section Date Perform the indicated multiplication. Then simplify each radical expression ANSWERS m x a b x 16x a 115a (1)(517) 22. (2 15)( 111)
6 ANSWERS (15)(2 110) (12 15) 26. (41)(16) 1(15 1) (215 1) (21 17) (1 5)(1 ) 0. (15 2)(15 1) (15 1)(15 ) 2.. (15 2)(15 2) (110 5)(110 5) 6. (12 )(12 7) (17 5)(17 5) (111 )(111 ) (1x )(1x ) 8. (1a 4)(1a 4) (1 2) (15 ) (1y 5) (1x 4) Perform the indicated division. Rationalize the denominator if necessary. Then simplify each radical expression a m
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