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Math Help - Properties of Roots and Absolute Values

  1. #1
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    Unhappy Properties of Roots and Absolute Values

    Hi. I'm in the initial review section of a pre-calc book and a little confused about something. The problem is to rationalize the denominator of the following expression:

    \frac{1}{\sqrt[4]{y^3}}

    The answer the book gives is

    \frac{\sqrt[4]{y}}{y}

    However, when I work through the problem I end up with:

    \frac{\sqrt[4]{y}}{|y|}

    Because of the properties of nth Roots (given in the book) which indicate that

    \sqrt[n]{a^n} = a\ \mbox{if n is odd}

    but

    \sqrt[n]{a^n} = |a|\ \mbox{if n is even}

    Have I missed something, or did the book make a mistake? I suppose it is a small point, but I keep getting confused about the issue of roots and absolute values.
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  2. #2
    Super Member TheChaz's Avatar
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    Re: Properties of Roots and Absolute Values

    You can usually determine the correct answer by comparing when (for instance) y = -2 and y = 2.
    If y = -2, then the fourth root (of -8) is imaginary. So there must be an assumption that y is positive in this example.
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