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College Algebra Exam Review 79

# College Algebra Exam Review 79 - 89 2.1 FIRST RESULTS but...

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2.1. FIRST RESULTS 89 but by the associative law, the first two and the last two are equal. Thus there are at most three different product of four elements: a.bcd/; .ab/.cd/; .abc/d: Using the associative law, we see that all three are equal: a.bcd/ D a.b.cd// D .ab/.cd/ D ..ab/c/d D .abc/d: Thus there is a well-defined product of four elements, which is independent of the way the elements are grouped for multiplication. There are 14 ways to group five elements for multiplication; we won’t bother to list them. Because there is a well-defined product of four or less elements, independent of the way the elements are grouped for multiplica- tion, there are at most four distinct products of five elements: a.bcde/; .ab/.cde/ .abc/.de/; .abcd/e: Using the associative law, we can show that all four products are equal, a.bcde/ D a.b.cde// D .ab/.cde/; etc. Thus the product of five elements at a time is well-defined, and inde- pendent of the way that the elements are grouped for multiplication.
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