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Thread: single algebraic fraction

  1. #1
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    single algebraic fraction

    Write each expression as a single algebraic fraction.
    (1-xy^-1)^-1
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  2. #2
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    $\displaystyle (1-(xy)^{-1})^{-1}$

    Remember:
    $\displaystyle a^{-1} = \frac{1}{a}$

    So:
    $\displaystyle (1-(xy)^{-1})^{-1} = \frac{1}{1-(xy)^{-1}} = \frac{1}{1-\frac{1}{xy}}$

    Combine 1 and $\displaystyle \frac{1}{xy}$. Remember that $\displaystyle 1 = \frac{xy}{xy}$.

    $\displaystyle \frac{1}{1-\frac{1}{xy}} = \frac{1}{\frac{xy}{xy}-\frac{1}{xy}} = \frac{1}{\frac{xy - 1}{xy}}$

    Now, this expression can be rearranged like this:

    $\displaystyle \frac{1}{\frac{xy - 1}{xy}} = 1 \cdot \frac{xy}{xy-1} = \frac{xy}{xy-1}$
    Last edited by Chop Suey; Aug 11th 2008 at 01:30 PM.
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  3. #3
    Super Member wingless's Avatar
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    $\displaystyle xy^{-1}$ can mean $\displaystyle \frac{x}{y}$ too.
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  4. #4
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    If you meant $\displaystyle xy^{-1}$ as Wingless pointed out:

    $\displaystyle (1-xy^{-1})^{-1}$

    $\displaystyle = \frac{1}{1-\frac{x}{y}}$

    $\displaystyle = \frac{1}{\frac{y-x}{y}}$

    $\displaystyle = 1 \cdot \frac{y}{y-x} = \frac{y}{y-x}$
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