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6.3 Combining and Simplifying Radicals that Contain Variables BobsMathClass.Com Copyright © 2010 All Rights Reserved. 1 Your Turn Problem #1 Combining Radicals Combining Radicals is very similar to combining like terms. In both cases, you end up adding coefficients. For example, We may only combine (add or subtract) radicals if the radicands are the same and the indexes are the same.

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6.3 Combining and Simplifying Radicals that Contain Variables BobsMathClass.Com Copyright © 2010 All Rights Reserved. 2 (Now that we have like radicals, we can combine them.) We can not combine these radicals in current form because the radicands are different. However, we should always start by writing the each radical expression in simplest radical form. Answers: Your Turn Problem #2

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6.3 Combining and Simplifying Radicals that Contain Variables BobsMathClass.Com Copyright © 2010 All Rights Reserved. 3 Answer: We need to write each radical in simplest radical form. For cubes, circle in groups of 3. Answer: Answer Your Turn Problem #3

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6.3 Combining and Simplifying Radicals that Contain Variables BobsMathClass.Com Copyright © 2010 All Rights Reserved. 4 Simplifying Square Roots That Contain Variables As always, there is usually more than one approach to covering a topic. We had two methods (choices) for simplifying square roots in the last section. Recall the methods with the following example. Observe: The square root of a variable with an even exponent equals the variable with an exponent of half of the exponent in the radicand. Square Roots of Variables that are Perfect Squares Next Slide

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6.3 Combining and Simplifying Radicals that Contain Variables BobsMathClass.Com Copyright © 2010 All Rights Reserved. 5 Examples: We can rewrite a square root as a product where one of the radicals is a perfect square root (i.e. an even exponent). Square Roots of Variables that are not Perfect Squares Another approach: 1. Recall the index # of a square root is 2. Write in the index #. 2.Divide the index # into the exponent of the variable in the radicand. The quotient will be the exponent of the variable written in front. The remainder will be the exponent of the variable inside the radical. Note: To avoid a technical discussion about negative radicands under even-indexed radicals (non-real numbers), we are going to assume that all variables represent positive numbers.

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6.3 Combining and Simplifying Radicals that Contain Variables BobsMathClass.Com Copyright © 2010 All Rights Reserved. 6 Example 4. Simplify the following: Answers: Remember, divide the exponent inside by 2. This answer will be the exponent of the variable in front of the radical. If there is a remainder, this will be the exponent inside. Answers: Your Turn Problem #4 Simplify the following:

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6.3 Combining and Simplifying Radicals that Contain Variables BobsMathClass.Com Copyright © 2010 All Rights Reserved. 7 Simplifying Square Roots where the Radicand has both Numbers and Variables. The idea here is to combine the two types of radical expressions and write in simplest radical form. First work on the number using either method. Then work on the variables. Example 5. Simplify the following: Solutions: Answers: Your Turn Problem #5 Simplify the following:

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6.3 Combining and Simplifying Radicals that Contain Variables BobsMathClass.Com Copyright © 2010 All Rights Reserved. 8 Example 6. Simplify the following: Solutions: Your Turn Problem #6 Simplify the following: Answers:

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6.3 Combining and Simplifying Radicals that Contain Variables BobsMathClass.Com Copyright © 2010 All Rights Reserved. 9 Observe: The exponent in the answer equals the exponent of the radicand divided by 3. Cube Roots of Variables that are Perfect Cubes Examples: We can rewrite the cube root as a product where one of the radicals is a perfect cube root (i.e. the exponent is divisible by 3). Cube Roots of Variables that are Not Perfect Cubes Another approach: Divide the index # into the exponent of the variable in the radicand. The quotient will be the exponent of the variable written in front of the radical sign. The remainder will be the exponent of the variable inside the radical. Next Slide

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6.3 Combining and Simplifying Radicals that Contain Variables BobsMathClass.Com Copyright © 2010 All Rights Reserved. 10 Remember, divide the exponent inside by 3. This answer will be the exponent of the variable in front of the radical. If there is a remainder, this will be the exponent of the variable inside. Remember to write the index # on the radical in your final answer. Answers: Example 7. Simplify the following. Your Turn Problem #7 Simplify the following: Answers:

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6.3 Combining and Simplifying Radicals that Contain Variables BobsMathClass.Com Copyright © 2010 All Rights Reserved. 11 Simplifying Cube Roots where the Radicand has both Numbers and Variables. Again, the idea here is to combine the two types of radical expressions and write in simplest radical form. First work on the cube root of the number using either method. Then work on the variables. Example 8. Simplify the following:Answers: Your Turn Problem #8 Simplify the following: Answers: The End B.R. 10-18-2006

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