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8 Pages

### 7.3A

Course: MATH 1825, Fall 2008
School: Michigan State University
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Word Count: 635

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A. List of Squares/Cubes/Etc. To Memorize 5. Higher Powers 4. Fifth Powers 3. Fourth Powers 2. Cubes 1. Squares 7.3A Simplifying Roots and Radicals 1 B. Simplifying Roots To simplify a root, we remove the largest perfect power from inside the root and...

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A. List of Squares/Cubes/Etc. To Memorize 5. Higher Powers 4. Fifth Powers 3. Fourth Powers 2. Cubes 1. Squares 7.3A Simplifying Roots and Radicals 1 B. Simplifying Roots To simplify a root, we remove the largest perfect power from inside the root and put it as the base outside. Note: If you don't remove the largest, your root may not be fully simplified! Example 1: Solution We identify the largest number in the cube table that goes into Since Since Ans , the comes out as a . Thus, we have Solution We identify the largest number in the fourth power table that goes into Since Since Ans , the comes out as a . Thus, we have 2 is the largest number to go into , we write Example 2: Simplify . is the largest number to go into , we write Simplify . . . Example 3: Solution Simplify We identify the largest number in the square table that goes into Since Since Ans is the largest number to go into , the , we write comes out as a . Thus, we have C. Simplifying Radicals 1. First simplify the coefficient root. 2. To simplify the variables, use the "divide and remainder" trick: divide the power by the index; the quotient is the power that comes out and the remainder is the power that stays in. Note: For simplicity, we will assume during the next few sections that all variables represent positive real numbers so that we don't have absolute value issues. See the comments at the end. 3 . . D. Examples Example 1: Simplify . Assume that all variables represent positive real numbers. Solution 1. First simplify the coefficient root: The largest fourth power that goes into Thus, we have is 2. Now simplify the variables (divide and remainder trick): For , For , For , R R R Ans Thus we have 4 . . Example 2: Simplify . Assume that all variables represent positive real numbers. Solution 1. First simplify the coefficient root: The largest cube that goes into Thus, we have is 2. Now simplify the variables (divide and remainder trick): For , For , , For R R R For , R Example 3: Simplify . Assume that all variables represent positive real numbers. Solution 1. First simplify the coefficient root: The largest square that goes into is Ans 5 Thus we have . . . Thus, we have 2. Now simplify the variables (divide and remainder trick): For , For , For , R R R Thus we have Ans E. Comments 1. An alternate way to simplify roots is to write the prime factorization of the number, and then do the "divide and remainder" trick like what is used for variables. This procedure is longer, but useful if you can't figure out how to break down the root. For example, to find : Now use the divide and remainder trick: For , we have For , we have 6 Thus R R . 2. If you use the alternate method to find roots, it doesn't matter "how" you factor the number into prime powers. It is a fact that the final factorization (up to order) must be the same no matter how you do it. This fact is called the Fundamental Theorem of Arithmetic. 3. The justification for the simplification method given in this section comes from the &qu...

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