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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern