SOLUTION: Use De Moivre's Formula to find (-1+2i)^36

To find (x+yi)n
First change x+yi to trigonometric form.

To change x + iy to trigonometric form, we plot
the point (x,y), and draw a right triangle with
vertices (x,y), (x,0), and (0,0)
We calculate the hypotenuse r from r² = x² + y²,
which is called the modulus or absolute value
of the complex number x+yi
We calculate the angle q
from either

tanq = ,
or 
sinq = ,
or
cosq = ,

Then x + yi has the trigonometric form:

x + yi = r(cosq + i·sinq)

Then we use the formula:

[r(cosq + i·sinq)]n = rn[cos(nq) + i·sin(nq)]

The real part of -1+2i is -1
The imaginary part of -1+2i is 2, the coefficient of i.

We plot the point (-1,2), and draw a right triangle with
vertices (-1,2), (-1,0), and (0,0)



Next we calculate r:

r² = x² + y²
r² = (-1)² + 2²
r² = 1 + 4
r² = 5
r = 



Now we need to calculate the angle q
indicated by the arc:



We first calculate its reference angle from, say

tan(refq) = ,

So we find the reference angle from the inverse tangent
on the calculator:

refq = 63.4349°

Now we know that q is in 
Quadrant II, so we subtract the reference angle from 
180° and get q = 116.5651°

So the trigonometric form is
           
-1 + 2i = [cos(116.5651°)+ i·sin(116.5651°)]

So using the formula

[r(cosq + i·sinq)]n = rn[cos(nq) + i·sin(nq)]

[(cos116.5651° + i·sin116.5651°)36 =
[cos(36·116.5651°) + i·sin(36·116.5651°)] = 
[cos(4196.3436°) + i·sin(4196.3436°)] = 
[cos(4196.3436°) + i·sin(4196.3436°)]

Now we can replace the angle 4196.3436° by its smallest 
positive coterminal angle.

To do this we divide the angle by 360° and get 11.65651.  The
whole part is 11 so we subtract 360° 11 times, which means we 
subtract 11·360° or 3960° from 4196.3436° and get

4196.3436° - 3960° = 236.3436°

So the answer in trig form is

518(cos236.3436° + i·sin236.3436°)

Using a calculator we get

-2.114 - 3.175i for the answer in standard form.

Edwin