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Math Help - Simplifying with fractional exponets

  1. #1
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    Simplifying with fractional exponets

    Simplify the expressions.

    Problem 1:
    \frac{2(1+x)^{1/2} - x(1+x)^{-1/2}}{x+1}

    The final answer to problem 1 is:
    \frac{x+2}{(x+1)^{3/2}}

    Problem 2:
    \frac{(7-3x)^{1/2} + \frac{3}{2} x (7-3x)^{-1/2}}{7-3x}
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  2. #2
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    Re: Simplifying with fractional exponets

    Hello, friesr!

    \text{Simplify:}

    . . [1]\;\;\frac{2(x+1)^{\frac{1}{2}} - x(x+1)^{-\frac{1}{2}}}{x+1}

    Multiply by \frac{(x+1)^{\frac{1}{2}}}{(x+1)^{\frac{1}{2}}}

    . . \frac{(x+1)^{\frac{1}{2}}}{(x+1)^{\frac{1}{2}}} \cdot \frac{2(x+1)^{\frac{1}{2}} - x(x+1)^{-\frac{1}{2}}}{x+1}

    . . =\;\frac{2(x+1)-x}{(x+1)^{\frac{3}{2}}} \;=\;\frac{x+2}{(x+1)^{\frac{3}{2}}}




    [2]\;\frac{(7-3x)^{1/2} + \frac{3}{2} x (7-3x)^{-1/2}}{7-3x}

    Multiply by \frac{2(7-3x)^{\frac{1}{2}}}{2(7-3x)^{\frac{1}{2}}}

    . . \frac{2(7-3x)^{\frac{1}{2}}}{2(7-3x)^{\frac{1}{2}}}\cdot \frac{(7-3x)^{\frac{1}{2}} + \frac{3}{2}x(7-3x)^{-\frac{1}{2}}}{7-3x}

    . . =\;\frac{2(7-3x) + 3x}{2(7-3x)^{\frac{3}{2}}} \;=\; \frac{14- 6x + 3x}{2(7-3x)^{\frac{3}{2}}} \;=\;\frac{14 - 3x}{2(7-3x)^{\frac{3}{2}}}

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  3. #3
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    Re: Simplifying with fractional exponets

    Seeing how this works but not sure on what to choose to multiply thru by.

    On this problem:

    \frac{2x(x+6)^{1/2} - x^{2}(4)(x+6)^{3}}{(x+6)^8}

    \frac{(x+6)^{1/2}}{(x+6)^{1/2}} * \frac{2x(x+6)^{1/2} - x^{2}(4)(x+6)^{3}}{(x+6)^8}

    \frac{2x(x+6) - x^{2}(4)(x+6)^{7/2}}{8(x+6)^{3/2}}

    \frac{x(x+6) - x^{2}(2)(x+6)^{7/2}}{4(x+6)^{3/2}}

    It does not look like my choice is working out well.
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  4. #4
    MHF Contributor Siron's Avatar
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    Re: Simplifying with fractional exponets

    You have:
    \frac{2x(x+6)^{\frac{1}{2}}-4x^2(x+6)^3}{(x+6)^8}
    =\frac{(x+6)^{\frac{1}{2}}[2x-4x^2(x+6)^{\frac{5}{2}}]}{(x+6)^{\frac{1}{2}}(x+6)^{\frac{15}{2}}}
    =\frac{2x-4x^2(x+6)^{\frac{5}{2}}}{(x+6)^{\frac{15}{2}}}
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  5. #5
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    Re: Simplifying with fractional exponets

    \frac{2x - 4x^{2}(x+6)^{5/2}}{(x+6)^{15/2}}

    We can factor out a 2x

    \frac{2x (1 - 2x(x+6)^{5/2}}{(x+6)^{15/2}}

    Can we factor out more (x+6)?
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  6. #6
    Member sbhatnagar's Avatar
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    Re: Simplifying with fractional exponets

    Quote Originally Posted by friesr View Post
    \frac{2x - 4x^{2}(x+6)^{5/2}}{(x+6)^{15/2}}

    We can factor out a 2x

    \frac{2x (1 - 2x(x+6)^{5/2}}{(x+6)^{15/2}}

    Can we factor out more (x+6)?
    You can factor out (x+6) but I don't think it would be useful.

    \begin{align*}\frac{2x - 4x^{2}(x+6)^{5/2}}{(x+6)^{15/2}} &=\frac{2x}{(x+6)^{15/2}} - \frac{4x^{2}(x+6)^{5/2}}{(x+6)^{15/2}} \\ &=\frac{2x}{(x+6)^{15/2}} - \frac{4x^{2}}{(x+6)^{5}}\end{align*}

    This is as far as I can simplify.

    Please see this link:simplify (2x-4x^2(x+6&... - Wolfram|Alpha
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