Section 4: Radical Expressions

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

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Section 9.4 Radical Expressions 925 Version: Fall 2007 9.4 Radical Expressions In the previous two sections, we learned how to multiply and divide square roots. Specifically, we are now armed with the following two properties. Property 1. Let a and b be any two real nonnegative numbers. Then, a b = ab, and, provided b = 0 , a b = a b . In this section, we will simplify a number of more extensive expressions containing square roots, particularly those that are fundamental to your work in future mathe- matics courses. Let’s begin by building some fundamental skills. The Associative Property We recall the associative property of multiplication. Associative Property of Multiplication. Let a , b , and c be any real numbers. The associative property of multiplication states that ( ab ) c = a ( bc ) . (2) Note that the order of the numbers on each side of equation (2) has not changed. The numbers on each side of the equation are in the order a , b , and then c . However, the grouping has changed. On the left, the parentheses around the product of a and b instruct us to perform that product first, then multiply the result by c . On the right, the grouping is different; the parentheses around b and c instruct us to perform that product first, then multiply by a . The key point to understand is the fact that the different groupings make no difference. We get the same answer in either case. For example, consider the product 2 · 3 · 4 . If we multiply 2 and 3 first, then multiply the result by 4, we get (2 · 3) · 4 = 6 · 4 = 24 . On the other hand, if we multiply 3 and 4 first, then multiply the result by 2, we get 2 · (3 · 4) = 2 · 12 = 24 . Copyrighted material. See: 1
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926 Chapter 9 Radical Functions Version: Fall 2007 Note that we get the same result in either case. That is, (2 · 3) · 4 = 2 · (3 · 4) . The associative property, seemingly trivial, takes on an extra level of sophistication if we apply it to expressions containing radicals. Let’s look at an example. l⚏ Example 3. Simplify the expression 3(2 5) . Place your answer in simple radical form. Currently, the parentheses around 2 and 5 require that we multiply those two numbers first. However, the associative property of multiplication allows us to regroup, placing the parentheses around 3 and 2, multiplying those two numbers first, then multiplying the result by 5 . We arrange the work as follows. 3(2 5) = (3 · 2) 5 = 6 5 . Readers should note the similarity to a very familiar manipulation. 3(2 x ) = (3 · 2) x = 6 x In practice, when we became confident with this regrouping, we began to skip the intermediate step and simply state that 3(2 x ) = 6 x . In a similar vein, once you become confident with regrouping, you should simply state that 3(2 5) = 6 5 . If called upon to explain your answer, you must be ready to explain how you regrouped according to the associative property of multiplication. Similarly, 4(5 7) = 20 7 , 12(5 11) = 60 11 , and 5( 3 3) = 15 3 .
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