complex numbers multivalued function

(j^1/2)^3 and (j^3)^1/2

calculate these and hence find a condition such that (j^1/m)^n=(j^n)^1/m


i calculated this first one right but i got the same answer for the second as the first but theyre supposed to be different :/

and also then i have no idea how to find this condition after that. anyone got any ideas on the condition and could you explain it please?

Is this an A level course you're learning from? What does your textbook say about the allowable range for the argument of a complex number?

yes, it is -pi to pi to give the principal polar compex number,

OK, so I suspect you've got the first one correct - j is vertically upwards (arg pi/2), so raised to the power 1/2 gives you a number with arg pi/4, then cube it to get a number with arg 3pi/4.

However, j^3 = -j points vertically downwards so its argument = -pi/2. So what must the argument be when you take the 1/2 power, and where does the result lie in that case?

-pi/4 and 4th quadrant?

That's what I reckon too, so one of your answers should be in the 2nd quadrant and the other in the 4th quadrant - hence the difference.

ahh ok cheers i tried again and got it now.

how would i decide the condition though; i am still unsure on that!