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Find the 3rd Root of equation $z= 8\left(\cos(\frac \pi 4) + i \sin(\frac \pi 4)\right)$

If I write the values of cos and sin then I would have only one root, how to find 3 roots for above equation ?

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  • $\begingroup$ That quation has only one root. $z=8 e^{i\pi/4}$. $\endgroup$ – Pedro Tamaroff Nov 11 '14 at 15:39
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This equation has only one solution for $z$ (the given one). This one can also be written as $z = 8 e^{i\pi/4}$.

Perhaps you might want to find the 3rd root of $z$?

Then youre gonna have the solutions

$$"\sqrt[3]{z}" = \sqrt[3]{8} e^{i\pi/12} e^{i2 \pi k/ 3} = 2 e^{i2\pi(1/24+k/3)}$$ for $k = 0,1,2$

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  • $\begingroup$ what would be the 3rd root ? How to do that ? $\endgroup$ – pratik Nov 11 '14 at 15:41
  • $\begingroup$ So it would have 3 roots for k = 0, 1 and 2 $\endgroup$ – pratik Nov 11 '14 at 15:43
  • $\begingroup$ When multiplying complex numbers, the length/modulus will be multiplied and the angle (also called argument) will be added. So the third roots are those numbers with modulus wich is the third root of the original number, and the angle is divided by 3. But there are 3 solutions for roots, because if you multiply by $e^{i2\pi/3}$ you get another solution. (And another one by multiplying twice with this number.) $\endgroup$ – flawr Nov 11 '14 at 15:44

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