# Complex power raised over real number.

1. Aug 22, 2012

### PrashntS

1. I actually dont know if such kind of operation is even allowed.

A friend of mine raised this question, that can we raise a complex power over a real number. I solved it this way. Is this correct?

http://i45.tinypic.com/254vwux.jpg
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Last edited by a moderator: Aug 22, 2012
2. Aug 22, 2012

### SammyS

Staff Emeritus
Yes, that's fine.

3. Aug 22, 2012

### Millennial

Yes, you did it correctly. I am not quite sure approximating the transcandental functions with decimals, but I suppose that's fine.

4. Aug 22, 2012

### Ray Vickson

Your method is OK, but if you plan to use the results in further numerical computations, you should keep more digits of accuracy; nowadays, in this computer age, there is no barrier to retaining more digits. For example, it might be better (depending on future uses) to write
$$3^{5i} = e^{i 5 \ln 3 } = \cos(5 \ln 3) + i \sin(5 \ln 3) \doteq 0.7037573 - 0.7104404 i\, .$$

RGV

5. Aug 22, 2012

### PrashntS

Yeah this is obviously much better, I was just looking whether it is even possible or not as none of the books I use has such question.
Would be awesome if you could suggest some good book for complex number (pre collage).

6. Aug 23, 2012

### Bacle2

Are you also taking into account that there are infinitely-many solutions depending