<s>[INST] Find : $\sqrt[6]{\frac{\sqrt{2}+(-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i)^7}{(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i)^{11}}}$ in its algebraic form. Find : $$\sqrt[6]{\frac{\sqrt{2}+(-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i)^7}{(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i)^{11}}}$$ in its algebraic form.
Now, I kinda think it would not be wise to try to expand this, but rather apply de Moivre formula on the complex number in the numerator and denominator, then simplify that complex number within the root, and once again apply moivres formula.
I have tried but then I get the expression:$\sqrt[6]{\frac{\sqrt{2}+\cos{\frac{21 \pi}{4}+i\sin{\frac{21 \pi }{4}}}}{\cos \frac{11\pi}{4}+i\sin\frac{11 \pi}{4}}}$ and don't know what to do with it.
 [/INST] The binomial quantities being raised to powers are easy to simplify: $$\sqrt[6]{\frac{\sqrt{2}-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i}{-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i}}$$
Now it's quite easy to continue. $$\sqrt[6]{\frac{\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i}{-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i}}=\sqrt[6]{-1}$$
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