Thread: How do you calculate the nth root?

1. 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?

2. Re: How do you calculate the nth root?

Originally Posted by Lancet
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 $\displaystyle x^4-1=(x-1)(x+1)(x-\mathbf{i})(x+\mathbf{i})$
There are the four roots.

3. 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

4. Re: How do you calculate the nth root?

Originally Posted by Plato
We could factor $\displaystyle x^4-1=(x-1)(x+1)(x-\mathbf{i})(x+\mathbf{i})$
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:

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

Is that would I would use?

5. Re: How do you calculate the nth root?

Originally Posted by Lancet
recall a formula when working with complex numbers:
$\displaystyle |x^{1/n}|*[cos(\frac{\theta}{n} + \frac{2k\pi}{n}) + jsin(\frac{\theta}{n} + \frac{2k\pi}{n})]$
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 $\displaystyle n^{th}$ roots of $\displaystyle z$.
First find $\displaystyle \theta=\text{Arg}(z)$ then $\displaystyle r=\sqrt[n]{{\left| z \right|}}$.
The n roots are $\displaystyle r\exp \left( {\frac{{\theta + 2k\pi }}{n}\mathbif{i}} \right)~k=0,1,\cdots,n-1$

6. Re: How do you calculate the nth root?

Originally Posted by Plato

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.