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

Answer:

z1 = 7.71 + 0.02 i

z2 = 7.73 + 0.306 i

z3 = 7.78 + 0.59 i

Step-by-step explanation:

To find the roots you use:

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

n: the order of the roots

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

First, you write z in polar notation:

$z=re^{i\theta}\\\\r=\sqrt{(473)^2+(4)^2}=473.01\\\\\theta=tan^{-1}(\frac{4}{473})=0.48\°$

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

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

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