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Math Help - What is the Square of the Square Root of a Polynomial

  1. #1
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    What is the Square of the Square Root of a Polynomial

    Example: (\sqrt{1-x^2})^2

    Inuitively, I would say it's simply 1-x^2. Alternatively (\sqrt{1-x^2})(\sqrt{1-x^2}) appears to give \sqrt{1-2x^2+4x^2} which also does not seem correct.

    The question arises from this problem:

    \sqrt{1+(\frac{x}{\sqrt{1-x^2}})^2} where I constantly get 1 by stating \sqrt{1-x^2} = 1-x^2. The correct answer, according to my text book anyway, is \frac{1}{\sqrt{1-x^2}}.

    Thanks -
    Last edited by TaylorM0192; January 21st 2010 at 09:47 AM.
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  2. #2
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    Nevermind, I completely forgot that I needed to multiply the denominator by \sqrt{1-x^2} to keep the expression equal. The numerator as I simplified earlier, is 1 and there is nothing to simplify in the denominator => \frac{1}{\sqrt{1-x^2}}
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  3. #3
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    Quote Originally Posted by TaylorM0192 View Post
    Example: (\sqrt{1-x^2})^2

    Inuitively, I would say it's simply 1-x^2. Alternatively (\sqrt{1-x^2})(\sqrt{1-x^2}) appears to give \sqrt{1-2x^2+4x^2} which also does not seem correct.

    The question arises from this problem:

    \sqrt{1+(\frac{x}{\sqrt{1-x^2}})^2} where I constantly get 1 by stating \sqrt{1-x^2} = 1-x^2. The correct answer, according to my text book anyway, is \frac{1}{\sqrt{1-x^2}}.

    Thanks -
    Help me out:
    So you distributed the exponents to \frac {x}{\sqrt{1-x^2}} , how would you simplify ... ohh I see.

    So it simplifies to ..
    \sqrt {\frac {1}{1-x^2} + \frac {x^2}{1-x^2}}

    right? If so, then where did the x^2 in the numerator go?
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  4. #4
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    algebraic calculation

    √(1+((x/(√(1-x)))))
    =√(1+(((x)/(1-x))))
    =√(((1-x+x)/(1-x)))
    =√((1/(1-x)))
    =(1/(√(1-x)))
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