August 19, 2007
This session is based on the material in section 2.1 of Kelley's "The Humongous Book of Calculus Problems"
The section presents a series of problems involving simplifying expressions containing powers and square roots.
Problem 2.1

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Nope. Let's try this again, paying very close attention to the parentheses.

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Good. I must ensure that parentheses enclose both the entire numerator and the entire denominator.

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This is strange. The third term can be simplified further. Let's play with this for a moment.

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Fascinating. This does not appear to simplify to x.
There is another command called FullSimplify. Let's try that.

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Nope. That doesn't do it either.

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Okay. Mathematica is correct. This last expression does not simplify if x is a complex number. However if x is restricted to the real numbers, then we get the following:

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Perfect.

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Conclusion. Simplifying complex expressions involving powers sometimes fails to totally simplify the expression. I am not sure if this is a weakness with Mathematica or whether there is some subtle mathematics that says that such expressions do not simplify further. Note that Mathematica does not give a wrong answer - it simply fails to simplify the expression as far as one would expect.

Created by Mathematica  (August 19, 2007)