use the theorem below to

Question

 

Use the theorem below to find the indicated roots of the complex number. 

B. Represent each of the roots graphically. 

C. Express each of the roots in a standard form. 

For a positive integer n, the complex number z = r(costheta+ i sintheta ) has exactly n distinct nth roots given by… 

the nth root of r(cos(theta+2pik)/n+isin(theta+2pik)/n)) 

Where k = 0, 1, 2, … , n-1 

Find the fourth roots of 625i 

 

thank you!

Answer 1

Find the fourth roots of 625 i.

 

z = 625 i = 625 e^(i (pi/2 + 2n pi)), n integer.

 

z^(1/4) = 5 e^(i (pi/8 + n pi/2)), n integer.

 

n = 0, z^(1/4) = 5 e^(i (pi/8 ))

 

n = 1, z^(1/4) = 5 e^(i (pi/8 + pi/2)) = 5 e^(i 5pi/8)

 

n = 2, z^(1/4) = 5 e^(i (pi/8 + pi)) = 5 e^(i 9pi/8)

 

n = 3, z^(1/4) = 5 e^(i (pi/8 + 3 pi/2))= 5 e^(i 13pi/8)

 

n = 4, z^(1/4) = 5 e^(i (pi/8 + 2 pi)) = 5 e^(i (pi/8 ))

 

So there are 4 distinct fourth roots for n = 0,1,2,3.

 

They are equally spaced around a circle centered at the origin of the Argand Plane with radius 5.