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Math Help - Simplifying fractions involving surds and powers

  1. #1
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    Simplifying fractions involving surds and powers

    Hello, I intend to find my way round LaTex as soon as I can, but hope that this makes sense for now...

    Can someone advise me on how I can get from

    ((root(8.pi.n).((4.n)/e)^4n.(root(2.pi.n).(n/e)^n)^2)/(root(4.pi.n).((2.n)/e)^(2n))^3

    to

    (1/root(2)).2^(2n)

    ?

    I'm sure it's a case of manipulating surds etc, which I obviously need to practice!

    Thanks!
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  2. #2
    MHF Contributor

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    Re: Simplifying fractions involving surds and powers

    I think you mean \frac{\sqrt{8\pi n}\left(\frac{4n}{e}\right)^{4n}\sqrt{2\pi n}\left(\frac{n}{e}\right)^n}{\sqrt{4\pi n}\left(\left(\frac{2n}{e}\right)^{2n}\right)^3}

    I would say this is really a matter of converting to "exponent notation" and then using the "laws of exponents". In particular, you need to know that sqrt{x}= x^{1/2} so we can write that as
    \frac{(8\pi n)^{1/2}\left(\frac{4n}{e}\right)^{4n}\left(2\pi n\right)^{1/2}\left(4\pi n\right)^{1/2}\left(\frac{n}{e}\right)^n}}{\left(4\pi n\right)^{1/2}\left(\left(\frac{2n}{e}\right)^{2n}\right)^3}

    Now combine different powers of like bases. I see two different "like bases", 2\pi n to 1/2 powers and \frac{4n}{e}. Add the powers of those in the numerator, subtract the powers of those in the denominator.
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