What are the Powers of Complex Numbers?

[cepheid]

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

 

[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.

 

[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:

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

Is it?

 

[Curious3141]

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

 

[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|.

 

[cepheid]

This is a trick question, with a trivial solution.

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

Thanks for your help, everyone.

Halls:

Yeah, we also had to prove |z1z2| = |z1||z2| in this homework assignment
so I can see where those come from.

Thanks again.