Complex roots (review) (advise)

Question

I have to find the complex roots and want a review of my procedure to see if is correct

A. $$\sqrt{3i}$$ $$\left |z \right |=3 $$ $$phase= 90^{\circ}=\displaystyle\frac{\pi}{2}$$

$$3^{1/2}\left(\displaystyle\frac{\cos(\pi)+2k\pi}{4}+i\displaystyle\frac{\sin(\pi)+2k\pi}{4}\right)$$

B. $$\sqrt[3]{3+7i}$$ $$\left |z \right |=\sqrt{58} $$ $$phase= 1.1659$$

$$\sqrt{58}^{1/2}\left(\displaystyle\frac{\cos(1.1659)+2k\pi}{2}+i\displaystyle\frac{\sin(1.1659)+2k\pi}{2}\right)$$

C.$$x^4=7-3i$$ well, i don´t know what to do here

D. $$\sqrt[5]{-3}$$ $$\left |z \right |=3 $$ $$phase= 0$$

$$3^{1/5}\left(\displaystyle\frac{\cos(2k\pi)}{5}+i\displaystyle\frac{\sin(2k\pi)}{5}\right)$$

E.$$x^6-7=0$$ i don´t know what to do here either ...

Answer 1

Hints:

• if $z=a+ib$, then $|z|=\sqrt{a^2+b^2}$ (for example, you forgot the square root in A);
• $\cos(\frac{\pi}{4}+2k\pi)\neq\frac{\cos(\pi)+2k\pi}{4}$;
• for cases like C and E, you can first transform that into a problem like this $x^n=re^{i\theta}$ and proceed from there;
• Wolfram Alpha provides functions like "arg" and "abs" that compute an argument and the magnitude of a complex number; you can use tools like these to check your answers.