# Powers of Complex Numbers

1. Nov 30, 2009

### Phyisab****

1. The problem statement, all variables and given/known data

Obtain z^10 for z=-1+i

2. Relevant equations

z=re^i(theta)

3. The attempt at a solution

Theta is 3pi/4. So z^10 = 32e^i(15pi/2).

The answer in my book is 32e^i(3pi/2) but I'm pretty sure that's wrong, can anyone confirm?

2. Nov 30, 2009

### latentcorpse

yes but $e^{\frac{15 i \pi}{2}} = e^{\frac{12 i \pi}{2}} \cdot e^{\frac{3 i \pi}{2}}$

does that help?

3. Nov 30, 2009

### andylu224

z = -1 + i = sqrt(2) cis(3/4 pi)

z^10 = 2^5 cis (10 x 3/4 pi) {De Moivre's Theorem}
= 32 cis (15/2 pi)
= -32i

4. Dec 1, 2009

### Phyisab****

So I was right? Why did the answer appear in the book that way? Or is it wrong?

5. Dec 2, 2009

### andylu224

though technically the answers should be 32e^-i(1/2pi).

remember, the range of principal argument is 0 < theta < pi (above the x-axis) &
-pi < theta < 0 (below the x-axis). so whenever you get a value beyond these ranges you have to convert your large value into an equivalent but smaller value within the range.

So you start at the line to the right of the origin of the x-axis (which is theta = 0). one revolution around all the quadrants back to the start is 2 pi. you keep on going until you finish up to 15/2 pi. you should end up at the line to the bottom of the origin of the y-axis. that is 3/2 pi.

6. Dec 2, 2009

### HallsofIvy

Staff Emeritus
The point is that $12 i \pi= 6 (2 i \pi)$ so that $e^{12 i\pi}= (e^{2 i\pi})^6= 1$.

$e^{15 i \pi}= e^{3 i\pi}$ but it is best always to write the argument between 0 and $2\pi$.