Find all 4th roots of z=5000+i

Find all 4th roots of z=5000+i.  Show all work and draw a graph to reprecipent the answers.

I think I have worked out how to do this quite neatly I think but I do not know the correct terminology. 

I am not familiar with the correct terms so i used this page to help me there.

http://www.sparknotes.com/math/precalc/complexnumbers/terms.html

The distance (modulus) of this point from 0 is   \(\sqrt{5000^2+1^1}=\sqrt{25000001}\)

The distance (modulus) from 0 that the roots will be is  \((\sqrt{25000001})^{1/4}=25000001^{1/8}=\sqrt[8]{25000001}\)

(25000001)^(1/8) = 8.4089641945819655     approx 8.4090

First, 2pi/4 = pi/2  so the roots will be approx 8.409  units from 0 and pi/2 radians apart.

z=5000+i

I have just drawn a right angle triangle on a scrap of paper to work this out.

\(z=\sqrt{25000001}*(\frac{5000}{\sqrt{25000001}}+\frac{1i}{\sqrt{25000001}})\)

\(cos\theta=\frac{5000}{\sqrt{25000001}}\qquad and \qquad sin\theta=\frac{1}{\sqrt{25000001}}\)

argument (angle) of the first root is     acos(5000/sqrt(25000001) = 0.000199999998

asin(1/sqrt(25000001)) = 0.000199999997

They had to be the same, I was just showing you. :)

So the angle (argument) is very close to 0.0002 radians

The first 4th root angle is \(acos(\frac{5000}{\sqrt{25000001}})\div4\approx 0.0002\div4 = 0.00005 \;radians\)

So the angles of the 4 roots will be

0.00005, 0.00005+pi/2, 0.00005+pi, 0.0005+3pi/2

0.00005+pi/2 = 1.5708463267948966                    approx 1.5708 radians     

0.00005+pi/2+pi/2 = 3.1416426535897932            approx 3.1416  radians

0.00005+pi/2+pi/2+pi/2 = 4.7124389803846899    approx  4.7124  radians

So the 4th roots of  z=5000+i    are

8.409e^(0.00005i), 8.409e^(1.5708i), 8.409e^(3.1416i), 8.409e^(4.712i)

Check the first one.     

(8.409e^(0.00005i))^4 = 5000.08516065925605241+1.000017045882085632i   near enough

z = (5000+i)^(1/4)

Algebraic form:
z = 8.4089642+0.0004204i

Exponential form:
z = 8.4089642 × ei 0°10″

Trigonometric form:
z = 8.4089642 × (cos 0°10″ + i sin 0°10″)

Polar form:
r = |z| = 8.40896
φ = arg z = 0.00286° = 0°10″ = 2.0E-5π

All 4th roots are:

13^(1/4) 147929^(1/8) e^(1/4 i tan^(-1)(1/5000))=8.4090+0.0004 i  (principal root)

13^(1/4) 147929^(1/8) e^((i pi)/2+1/4 i tan^(-1)(1/5000))=-0.0004+8.4090 i

13^(1/4) 147929^(1/8) e^(i pi+1/4 i tan^(-1)(1/5000))=-8.4090-0.0004 i

13^(1/4) 147929^(1/8) e^((3 i pi)/2+1/4 i tan^(-1)(1/5000))=0.0004-8.4090 i