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Thread: approximation

  1. #1
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    approximation

    Write down the first three terms in the binomial expansion of $\displaystyle \sqrt[4]{1+\frac{x}{81}}$
    Here i got

    $\displaystyle 1+\frac{x}{324}-\frac{x^2}{69984}$

    Then deduce that $\displaystyle \sqrt[4]{80}=2.9907$ .. not sure with this part

    That should be the approximation sign , i am not sure how to do it with latex .
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  2. #2
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    Quote Originally Posted by thereddevils View Post
    Write down the first three terms in the binomial expansion of $\displaystyle \sqrt[4]{1+\frac{x}{81}}$
    Here i got

    $\displaystyle 1+\frac{x}{324}-\frac{x^2}{69984}$

    Then deduce that $\displaystyle \sqrt[4]{80}=2.9907$ .. not sure with this part

    That should be the approximation sign , i am not sure how to do it with latex .
    Note that $\displaystyle \sqrt[4]{1+\frac{x}{81}} = \sqrt[4]{\frac{81+x}{81}} = \frac{1}{3} \sqrt[4]{81+x}$ so substitute $\displaystyle x = -1$ into the approximate expansion of $\displaystyle \sqrt[4]{1+\frac{x}{81}}$ and then re-arrange to get the approximate value of $\displaystyle \sqrt[4]{80}$.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    Note that $\displaystyle \sqrt[4]{1+\frac{x}{81}} = \sqrt[4]{\frac{81+x}{81}} = \frac{1}{3} \sqrt[4]{81+x}$ so substitute $\displaystyle x = -1$ into the approximate expansion of $\displaystyle \sqrt[4]{1+\frac{x}{81}}$ and then re-arrange to get the approximate value of $\displaystyle \sqrt[4]{80}$.

    THank you .
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