I suggested only the single value (144, 7) for what I assume was also the reason Jer adduced, that no part of the expression should be an imaginary/complex term. This would mean that n should not be greater than 156.25, and hence 144 would be the unique solution. Since Charlie thinks otherwise, invoking the title "just math" in support, I suppose we must see what pcbouhid had in mind: the phrase "find all positive integers n..." suggests more than one, but it also suggests a finite number and not an infinitude of answers. Further, why should he have limited n to only positive integers unless he thought the problem needed that restriction? Only he can say. My days of formal math are decades behind me, since I first marveled at deMoivre's theorem.
With Charlie's interpretation, may I point out, we would not only need to have definition for the square root of a complex number, but also of the square root of the square root of a complex number if the second major factor had a real component less than 12.5. In that case we may not be able to dodge the problem by mere "cancellation" of the complexes.
I propose the original solution, however gross if may appear.