<s>[INST] Find the product of all values of $(1+i\sqrt 3)^{\frac{3}{4}}$. Find the product of all values of $(1+i\sqrt 3)^{\frac{3}{4}}$.
My try:
$(1+i\sqrt 3)^{\frac{3}{4}}=\exp (\frac{3}{4}(Log(1+i\sqrt 3)))=\exp(\frac{3}{4}(\log2+i\frac{\pi}{3}+2n\pi i))$.
I am kinda stuck on how to find the product of all values of  the above expression.
Can someone please help me out.
 [/INST] $$1+i\sqrt3=2e^{i\pi/3}$$
The fourth roots of this number have common magnitude $2^{1/4}$ and arguments $\pi/12,7\pi/12,-5\pi/12,-11\pi/12$. The $\frac34$-powers thus have common magnitude $2^{3/4}$ and arguments $\pi/4,7\pi/4\equiv-\pi/4,-5\pi/4\equiv3\pi/4,-11\pi/4\equiv-3\pi/4$. The producf of all the four possible values of $(1+i\sqrt3)^{3/4}$ therefore has magnitude equal to the product of the magnitudes of each possible value, which is $8$, and argument equal to the sum of the arguments of each possible value, which is $0$. The final answer is $8$.
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