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Math Help - How are these two items equal?

  1. #1
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    How are these two items equal?

    Can someone show me how

    Sqrt(x)/x

    is equal to 1/Sqrt(x)?

    I know I can flip the denominator to:
    1/X * Sqrt(x)

    But I still can't understand how it gets to 1/Sqrt(x).

    What am I missing
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  2. #2
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    \displaystyle \frac{\sqrt{x}}{x} = \frac{x^{1\over 2}}{x} \times \frac{x^{-1\over 2}}{x^{-1\over 2}}= \frac{1}{x^{1\over 2}}= \frac{1}{\sqrt{x}}
    Last edited by pickslides; August 31st 2010 at 08:50 PM. Reason: Bad Latex
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  3. #3
    Super Member Bacterius's Avatar
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    Alternatively, dividing each member of the fraction by \sqrt{x} (Pickslides multiplied by its inverse) :

    \displaystyle \frac{\left ( \frac{\sqrt{x}}{\sqrt{x}} \right )}{\left ( \frac{x}{\sqrt{x}} \right )} = \frac{1}{\sqrt{x}}
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  4. #4
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    Yet another way:
    \sqrt{x} is defined to be the number such that (\sqrt{x})^2= (\sqrt{x})(\sqrt{x})= x so the denominator can be written as (\sqrt{x})(\sqrt{x})

    \frac{\sqrt{x}}{x}= \frac{\sqrt{x}}{(\sqrt{x})(\sqrt{x})}= \frac{1}{\sqrt{x}}
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  5. #5
    MHF Contributor undefined's Avatar
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    Yet another way!

    \displaystyle \frac{\sqrt{x}}{x}=\frac{x^{\frac{1}{2}}}{x^1}=x^{  \frac{1}{2}-1}=x^{-\frac{1}{2}}=\frac{1}{\sqrt{x}}
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  6. #6
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    HallsofIvy's way is how I'd normally show it,
    however, it does depend on whether one is working with surds or indices,
    or various simplification techniques.

    \displaystyle\frac{\sqrt{x}}{x}=\frac{1}{\sqrt{x}}

    If they are equal, we will have RHS=LHS after cross-multiplying...

    as in \frac{1}{2}=\frac{2}{4}=\frac{3}{6}...\Rightarrow\ 1(4)=2(2),\ 1(6)=3(2),\ 2(6)=4(3)....

    hence

    \displaystyle\frac{\sqrt{x}}{x}=\frac{1}{\sqrt{x}}

    since

    \sqrt{x}\ \sqrt{x}=x(1)
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