Solving Complex Equations: A Refresher Course

[forty]

It's been a long time since I've had to do this stuff so bare with me!

Compute and graph the following:

(a) -15+i/4+2i (multiply by the conjugate and I got -2.9+1.7i

(b) (27^1/3)⁴ Now for some reason I have a feeling this has 3 solutions using complex numbers but can not figure them out, I've tried using polar coordinates and tried googling/wikipedia for help on this I'm just really stuck.

(c) 27^{(4 + 1/3)} I presume this can also have 3 solutions if i split it up into 27^1/3 times 27⁴

Any help on this greatly appreciated.

 

[jmb]

(a) -15+i/4+2i (multiply by the conjugate and I got -2.9+1.7i

That's correct.

(b) (27^1/3)⁴ Now for some reason I have a feeling this has 3 solutions using complex numbers but can not figure them out, I've tried using polar coordinates and tried googling/wikipedia for help on this I'm just really stuck.

To start, try considering the simpler $z=27^{1/3}$ and then writing $z$ out in exponential form...

Let me know if that hint isn't enough.

(c) 27^{(4 + 1/3)} I presume this can also have 3 solutions if i split it up into 27^1/3 times 27⁴

It sounds like once you have done (b) you should be able to do this.