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Math Help - Stuck on binomial expressions x.X

  1. #1
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    Question Stuck on binomial expressions x.X

    So I'm currently doing questions on binomial expressions, and I'm not sure as to how to do it after this >.> Any guidance? Thank you :]

    This is the question: If x - \frac{1}{x}= \frac{8}{3}, find the value of x^{3} - \frac{1}{x^{3}}.
    So this was what I tried to do:
    I thought that it meant triple (we are currently doing expressions related to \left ( a-b \right )^{3}= a^{3}-3a^{2}b + 3ab^{2}+b^{3} ^^)
    So this was what I did:

    \left ( x-\frac{1}{x} \right )\left ( x-\frac{1}{x} \right )\left ( x-\frac{1}{x} \right )

    And then I simply substituted as given in the question:

    =\left ( \frac{8}{3} \right )\left ( \frac{8}{3} \right )\left ( \frac{8}{3} \right )

    And multiplied, and got the answer:
    =\frac{512}{27}= 26\frac{18}{27}

    But I'm pretty sure that what I have done is completely wrong.. any help?
    Last edited by Lexadis; March 29th 2013 at 11:26 AM.
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    Re: Stuck on binomial expressions x.X

    Quote Originally Posted by Lexadis View Post
    This is the question: If x - \frac{1}{x}= \frac{8}{3}, find the value of x^{3} - \frac{1}{x^{3}}.

     \begin{align*}  \left(x-\frac{1}{x}\right)^3&=x^3-3x+\frac{3}{x}-\frac{1}{x^3}\\&=\left(x^3-\frac{1}{x^3}\right)-3\left(x-\frac{1}{x}\right) \end{align*}
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    Re: Stuck on binomial expressions x.X

    Quote Originally Posted by Plato View Post
     \begin{align*}  \left(x-\frac{1}{x}\right)^3&=x^3-3x+\frac{3}{x}-\frac{1}{x^3}\\&=\left(x^3-\frac{1}{x^3}\right)-3\left(x-\frac{1}{x}\right) \end{align*}
    Oh I didn't think of it that way ;D
    But then again, I'm confused as to how to proceed after that.
    = \left (x-\frac{1}{x}  \right )^{3} - 3 * \frac{8}{3}
    = x^{3}- \frac{1}{x^{3}} -8
    = ?
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  4. #4
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    Re: Stuck on binomial expressions x.X

    Lexadis
    it seems to me that you do not follow at all the instructions of Plato...
    JUst substitute ...x^3-1/x^3=(8/3)^3+3(8/3) finally you will get 728/27 i.e 26.9629
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    Re: Stuck on binomial expressions x.X

    Oops sorry, didn't realize that x.X So, the final result is.. like this?

    Quote Originally Posted by Plato View Post
     \begin{align*}  \left(x-\frac{1}{x}\right)^3&=x^3-3x+\frac{3}{x}-\frac{1}{x^3}\\&=\left(x^3-\frac{1}{x^3}\right)-3\left(x-\frac{1}{x}\right) \end{align*}
    ......................................... = \left ( \frac{8}{3} \right )^{3} - 3\left ( \frac{8}{3} \right )
    ......................................... = \frac{512}{27} - 8
    ......................................... = \frac{512 - 216}{27}
    ......................................... = \frac{296}{27}
    ......................................... = 10\frac{26}{27}
    Last edited by Lexadis; March 29th 2013 at 11:20 AM.
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    Re: Stuck on binomial expressions x.X

    Quote Originally Posted by Lexadis View Post
    Oops sorry, didn't realize that x.X So, the final result is.. like this?


    ......................................... = \left ( \frac{8}{3} \right )^{3} - 3\left ( \frac{8}{3} \right )
    ......................................... = \frac{512}{27} - 8
    ......................................... = \frac{512 - 216}{27}
    ......................................... = \frac{296}{27}
    ......................................... = 10\frac{26}{27}

    You missed a sign: = \left ( \frac{8}{3} \right )^{3} {\color{red}+}~ 3\left ( \frac{8}{3} \right )
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    Re: Stuck on binomial expressions x.X

    Quote Originally Posted by Plato View Post
    You missed a sign: = \left ( \frac{8}{3} \right )^{3} {\color{red}+}~ 3\left ( \frac{8}{3} \right )
    I'm sorry but I just don't understand how a + comes in between.. any explanation?
    Last edited by Lexadis; March 29th 2013 at 11:25 AM.
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    Re: Stuck on binomial expressions x.X

    Quote Originally Posted by Lexadis View Post
    I'm sorry but I just don't understand how a + comes in between.. any explanation?

    You have \left(x-\frac{1}{x}\right)^3=\left(x^3-\frac{1}{x^3}\right)-3\left(x-\frac{1}{x}\right)

    that gives \left(\frac{8}{3}\right)^3=\left(x^3-\frac{1}{x^3}\right){~\color{red}-}~3\left(\frac{8}{3}\right)

    Now solve for \left(x^3-\frac{1}{x^3}\right).
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  9. #9
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    Re: Stuck on binomial expressions x.X

    Quote Originally Posted by Plato View Post
    You have \left(x-\frac{1}{x}\right)^3=\left(x^3-\frac{1}{x^3}\right)-3\left(x-\frac{1}{x}\right)

    that gives \left(\frac{8}{3}\right)^3=\left(x^3-\frac{1}{x^3}\right){~\color{red}-}~3\left(\frac{8}{3}\right)

    Now solve for \left(x^3-\frac{1}{x^3}\right).
    Ooh I finally get it (so I think) so this was what I got:
    ................................... \left(\frac{8}{3}\right)^3=\left(x^3-\frac{1}{x^3}\right){~\color{red}-}~3\left(\frac{8}{3}\right)

    ................................... \frac{512}{27}= x^{3}- \frac{1}{x^{3}}-3\left ( \frac{8}{3} \right )

    ................................... \frac{512}{27}= x^{3}- \frac{1}{x^{3}}-8

    ................................... \frac{512}{27}+8= x^{3}- \frac{1}{x^{3}}

    ................................... x^{3}- \frac{1}{x^{3}}= \frac{512+216}{27}

    ................................... x^{3}- \frac{1}{x^{3}}= \frac{728}{27}

    ................................... x^{3}- \frac{1}{x^{3}}= 26\frac{26}{27}


    Correct?
    Last edited by Lexadis; March 29th 2013 at 12:42 PM.
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