# Thread: Find All Complex Roots

1. ## Find All Complex Roots

Find all complex roots. Leave your answers in polar form with the argument in degrees.

(1) The complex fourth roots of sqrt{3} - i.

(2) The complex firth roots of -i.

2. Originally Posted by magentarita
Find all complex roots. Leave your answers in polar form with the argument in degrees.

(1) The complex fourth roots of sqrt{3} - i.

(2) The complex firth roots of -i.
Reading this might help: Complex numbers : De Moivre's theorem : De Moivre's theorem and nth roots

You might also find this helpful: http://shawtlr.net/trig/trig_m11_slides.pdf

3. Originally Posted by magentarita
Find all complex roots. Leave your answers in polar form with the argument in degrees.

(1) The complex fourth roots of sqrt{3} - i.

(2) The complex firth roots of -i.
1)$\displaystyle \sqrt{3} - i = | \sqrt{3} - i | e^{i \arg (\sqrt{3} - i)} = 2 e^{-i\pi/6}$.

Therefore, the fourth roots are: $\displaystyle 2^{1/4} e^{-\pi i/24}, -2^{1/4} e^{-\pi i/24}, i2^{1/4}e^{-\pi i/24}, -i 2^{1/4} e^{-\pi i/24}$.

2)$\displaystyle -i = e^{-\pi i /2}$.

Therefore, the fifth roots are: $\displaystyle e^{-\pi i/10}, \zeta e^{-\pi i/10}, \zeta^2 e^{-\pi i/10}, \zeta^3 e^{-\pi i/10}, \zeta^4 e^{-\pi i/10}$.

Where $\displaystyle \zeta = e^{2\pi i/5}$

4. ## Thanks but.....

Originally Posted by ThePerfectHacker
1)$\displaystyle \sqrt{3} - i = | \sqrt{3} - i | e^{i \arg (\sqrt{3} - i)} = 2 e^{-i\pi/6}$.

Therefore, the fourth roots are: $\displaystyle 2^{1/4} e^{-\pi i/24}, -2^{1/4} e^{-\pi i/24}, i2^{1/4}e^{-\pi i/24}, -i 2^{1/4} e^{-\pi i/24}$.

2)$\displaystyle -i = e^{-\pi i /2}$.

Therefore, the fifth roots are: $\displaystyle e^{-\pi i/10}, \zeta e^{-\pi i/10}, \zeta^2 e^{-\pi i/10}, \zeta^3 e^{-\pi i/10}, \zeta^4 e^{-\pi i/10}$.

Where $\displaystyle \zeta = e^{2\pi i/5}$
Thanks but your reply is far beyond precalculus. All my questions come a precalculus textbook that I am studying in preparation for a state test.