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Math Help - Express the following by using 't'

  1. #1
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    Goyang, South Korea
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    Express the following by using 't'

    When

    t=\frac{{a^{2}+b^{2}}}{{a^{2}-b^{2}}}+<br />
\frac{{a^{2}-b^{2}}}{{a^{2}+b^{2}}}

    express the following by using 't'

    \frac{{a^{8}+b^{8}}}{{a^{8}-b^{8}}}+<br />
\frac{{a^{8}-b^{8}}}{{a^{8}+b^{8}}}

    -----------------------------------------------------

    So,, I got this problem and the first thing I did was factoring
    and I found the common denominator of 't'

    Then it turned out that

    t=\frac{{2(a^{4}+b^{4})}}{{a^{4}-b^{4}}}

    Also I could change 'the follwing' into this.

    \frac{{2(a^{16}+b^{16})}}{{a^{16}-b^{16}}}

    Next, I tried dividing 'the following' by 't' so that I can find out what I should multiply to 't' to make it as 'the following'.

    \frac{{(a^{16}+b^{16})}({a^{4}-b^{4})}}{({a^{16}-b^{16})}({a^{4}+b^{4})}}


    .... And now I'm stuck.
    I felt like I was getting closer to the answer but....

    Any help would be appreciated..
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  2. #2
    MHF Contributor red_dog's Avatar
    Joined
    Jun 2007
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    Medgidia, Romania
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    1
    We have t=\frac{2(a^4+b^4)}{a^4-b^4}

    a^4t-b^4t=2a^4+2b^4\Rightarrow\left(\frac{a}{b}\right)^  4=\frac{t-2}{t+2}\Rightarrow\left(\frac{a}{b}\right)^8=\left  (\frac{t-2}{t+2}\right)^2.

    Now, \frac{a^8+b^8}{a^8-b^8}+\frac{a^8-b^8}{a^8+b^8}=\frac{\left(\frac{a}{b}\right)^8+1}{  \left(\frac{a}{b}\right)^8-1}+\frac{\left(\frac{a}{b}\right)^8-1}{\left(\frac{a}{b}\right)^8+1}

    Replace \left(\frac{a}{b}\right)^8 with \left(\frac{t-2}{t+2}\right)^2
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