step 1: ignoring the negation and the imaginary number, find the fourth root of 20. Since it is an irrational number, I will represent it as "qart(20)" meaning the quartenary root of 20.
Step 2:
Now, we need to find the quartenary root of the -i part of it. To do this, translate the number of -i in to polar form.
Use Euler's equation:
exp(i*theta) = cos(theta) + i*sin(theta)
cos(theta) + i*sin(theta) = -i
therefore:
cos(theta) = 0
sin(theta) = -1
Which correspondes to theta = -Pi/2 radians
Therefore:
-i = exp(-i*Pi/2)
Step 3:
To take the fourth root of this, divide the exponent by 4, since 1/4 is the exponent of a quartenary root.
qart(-i) = exp(-i*Pi/8)
therefore our result is the following:
qart(-20*i) = qart(20)*exp(-i*Pi/8)
As decimals:
qart(-20*i) = 2.115 * exp(-0.3927 * i)
Step 4:
This is one of the roots. To find all four, we need to distribute values at the same radius from the origin, but spaced out in equal 90 degree segments
Those angles are
theta1 = -Pi/8
theta2 = -Pi/8 + Pi/2 = 3/8*Pi
theta3 = -Pi/8 + Pi = 7/8*Pi
theta4 = -Pi/8 - Pi/2 = -5/8*Pi
Our results are the following:
qart1(-20*i) = qart(20)*exp(-i*Pi/8)
qart2(-20*i) = qart(20)*exp(i*3*Pi/8)
qart3(-20*i) = qart(20)*exp(i*7*Pi/8)
qart4(-20*i) = qart(20)*exp(-i*5*Pi/8)
As decimals:
qart1(-20*i) = 2.115*exp(-0.3927*i)
qart2(-20*i) = 2.115*exp(1.1781*i)
qart3(-20*i) = 2.115*exp(2.7489*i)
qart4(-20*i) = 2.115*exp(-1.9635*i)
