Question

5. Find all three cube roots of the the complex number z = 473 + 4i, and plot

them in the complex plane.
Verify the identity​

Answer 1

z1 = 7.71 + 0.02 i

z2 = 7.73 + 0.306 i

z3 = 7.78 + 0.59 i

Explanation:

To find the roots you use:

`z^{(1)/(n)}=r^{(1)/(n)}[cos((\theta+2\pi k)/(n))+isin((\theta+2\pi k)/(n))]` ( 1 )

n: the order of the roots

k: 0,1,2,...,n-1

First, you write z in polar notation:

`z=re^(i\theta)\\\\r=√((473)^2+(4)^2)=473.01\\\\\theta=tan^(-1)((4)/(473))=0.48\°`

Thus, by using these values for the angle and r in the expression (1), you obtain:

`k=0\\\\z_1=(473.01)^(1/3)[cos((0.48+2\pi(0))/(3))+isin((0.48+2\pi(0))/(3))]\\\\z_1=7.79(0.99+i2.79*10^(-3))=7.71+i0.02\\\\z_2=7.79[cos((0.48+2\pi(1))/(3))+isin((0.48+2\pi(1))/(3))]\\\\z_2=7.73+i0.306\\\\z_3=7.79[cos((0.48+2\pi(2))/(3))+isin((0.48+2\pi(2))/(3))]\\\\z_3=7.78+i0.59`

hence, from the previous results you obtain:

z1 = 7.71 + 0.02 i

z2 = 7.73 + 0.306 i

z3 = 7.78 + 0.59 i

I attached and image of the plot