Section 2: Multiplication Properties of Radicals

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 9.2 Multiplication Properties of Radicals 885 Version: Fall 2007 9.2 Multiplication Properties of Radicals Recall that the equation x 2 = a , where a is a positive real number, has two solutions, as indicated in Figure 1 . x y y = x 2 y = a a a Figure 1. The equation x 2 = a , where a is a positive real number, has two solutions. Here are the key facts. Solutions of x 2 = a . If a is a positive real number, then: 1. The equation x 2 = a has two real solutions. 2. The notation a denotes the unique positive real solution. 3. The notation a denotes the unique negative real solution. Note the use of the word unique . When we say that a is the unique positive real solution, 2 we mean that it is the only one. There are no other positive real numbers that are solutions of x 2 = a . A similar statement holds for the unique negative solution. Thus, the equations x 2 = a and x 2 = b have unique positive solutions x = a and x = b , respectively, provided that a and b are positive real numbers. Furthermore, because they are solutions, they can be substituted into the equations x 2 = a and x 2 = b to produce the results ( a ) 2 = a and b 2 = b, respectively. Again, these results are dependent upon the fact that a and b are positive real numbers. Similarly, the equation Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1 Technically, the notation calls for a nonnegative real square root, so as to include the possibility 2 0 .
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886 Chapter 9 Radical Functions Version: Fall 2007 x 2 = ab has unique positive solution x = ab , provided a and b are positive numbers. However, note that a b 2 = ( a ) 2 b 2 = ab, making a b a second positive solution of x 2 = ab . However, because ab is the unique positive solution of x 2 = ab , this forces ab = a b. This discussion leads to the following property of radicals. Property 1. Let a and b be positive real numbers. Then, ab = a b. (2) This result can be used in two distinctly different ways. You can use the result to multiply two square roots, as in 7 5 = 35 . You can also use the result to factor, as in 35 = 5 7 . It is interesting to check this result on the calculator, as shown in Figure 2 . Figure 2. Checking the result 5 7 = 35 . Simple Radical Form In this section we introduce the concept of simple radical form , but let’s first start with a little story. Martha and David are studying together, working a homework problem from their textbook. Martha arrives at an answer of 32 , while David gets the result 2 8 . At first, David and Martha believe that their solutions are different numbers, but they’ve been mistaken before so they decide to compare decimal approximations of their results on their calculators. Martha’s result is shown in Figure 3 (a), while David’s is shown in Figure 3 (b).
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Section 9.2 Multiplication Properties of Radicals 887 Version: Fall 2007 (a) Martha’s result. (b) David’s result.
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