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Math Help - How do you calculate the nth root?

  1. #1
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    How do you calculate the nth root?

    If I have something like

    x^4 = 1

    ...which I know is a very simple example, how do I calculate all the roots?
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  2. #2
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    Re: How do you calculate the nth root?

    Quote Originally Posted by Lancet View Post
    If I have something like
    x^4 = 1
    ...which I know is a very simple example, how do I calculate all the roots?
    We could factor x^4-1=(x-1)(x+1)(x-\mathbf{i})(x+\mathbf{i})
    There are the four roots.
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  3. #3
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    Re: How do you calculate the nth root?

    We have x^4 - 1 = 0 => ( x^2 + 1 )(x^2 - 1 ) = 0

    gives x^2 = 1 and x^2 = -1

    so roots of equation are 1; -1; i; -i
    Last edited by Isomorphism; July 2nd 2011 at 11:53 AM. Reason: waqarhaider: Do not write in capital letters. It is considered as shouting and thus rude. Next time, you will be reported.
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  4. #4
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    Re: How do you calculate the nth root?

    Quote Originally Posted by Plato View Post
    We could factor x^4-1=(x-1)(x+1)(x-\mathbf{i})(x+\mathbf{i})
    There are the four roots.
    That works for something simple, but what happens when we get to something more complex?

    I think I recall a formula when working with complex numbers:

    |x^{1/n}|*[cos(\frac{\theta}{n} + \frac{2k\pi}{n}) + jsin(\frac{\theta}{n} + \frac{2k\pi}{n})]


    Is that would I would use?
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  5. #5
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    Re: How do you calculate the nth root?

    Quote Originally Posted by Lancet View Post
    recall a formula when working with complex numbers:
    |x^{1/n}|*[cos(\frac{\theta}{n} + \frac{2k\pi}{n}) + jsin(\frac{\theta}{n} + \frac{2k\pi}{n})]
    The way you put the question in the OP lead us to think you were asking a different level of question.

    Yes that formula works if you understand its parts.
    Finding the n^{th} roots of z.
    First find \theta=\text{Arg}(z) then r=\sqrt[n]{{\left| z \right|}}.
    The n roots are r\exp \left( {\frac{{\theta  + 2k\pi }}{n}\mathbif{i}} \right)~k=0,1,\cdots,n-1
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  6. #6
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    Re: How do you calculate the nth root?

    Quote Originally Posted by Plato View Post

    The way you put the question in the OP lead us to think you were asking a different level of question.

    At the time I had posted it, I had completely forgotten about this formula. At some point, I remembered using it for complex numbers, and needed to double check that it was useable for just real numbers.

    I also didn't know if I was overcomplicating things, and if there was an easier way to do it.
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