Homework Help: 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

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.