# Powers of Complex Numbers

1. Jan 15, 2005

### cepheid

Staff Emeritus
Hello,

I'm having trouble with this problem:

$$\left| \frac{(\pi + i)^{100}}{(\pi - i)^{100}} \right| = \ \ ?$$

My first thought was, "put it in polar form and simpify," but that is not helping.

For the numerator pi + i :
$$r = \sqrt{\pi^2 + 1}$$

$$\theta = \arctan{ \frac{1}{\pi} }$$ = ???

I don't see how this will help, it's not an easy one to put in polar form

I can also see that the numerator and denominator are complex conjugates, so maybe that is the starting point. But I can't see how to proceed

2. Jan 15, 2005

### Curious3141

Hint : they're only asking for the magnitude of the ratio of the two. Since, the numerator and denominator are complex conjugates, what can you say about their magnitudes ? Don't they cancel out ? What will be left when the magnitudes cancel out ?

This is a trick question, with a trivial solution.

3. Jan 15, 2005

### gnome

Sorry to break into your thread, but I suddenly have an irrepressible need to know if this is allowed in the world of complex numbers:

$$\left | \frac{(pi + i)^{100}}{(pi - i)^{100}}\right | { }^?_= \left ( \frac{|(pi + i)|}{|(pi - i)|} \right )^{100}$$

Is it?

4. Jan 15, 2005

### Curious3141

Yes, that works out. In fact, the numerator and demoninator can be any old complex numbers, not necessarily conjugates.

5. Jan 15, 2005

### HallsofIvy

Staff Emeritus
In other words, you don't NEED to know the argument. All you are asked about is the modulus so that's all you need to know!

In general, |zn|= |z|n and |a/b|= |a|/|b|.

6. Jan 15, 2005

### cepheid

Staff Emeritus
I can't believe I didn't see that! Even though I realised the numerator and denominator were conjugates.