<s>[INST] Prove: $(\frac{1+i\sqrt{7}}{2})^4+(\frac{1-i\sqrt{7}}{2})^4=1$ $$\left(\frac{1+i\sqrt{7}}{2}\right)^4+\left(\frac{1-i\sqrt{7}}{2}\right)^4=1$$
I tried moving the left exponent to the RHS to then make difference of squares exp. $(x^2)^2$. Didn't get the same on both sides though. Any help?
 [/INST] We need to solve for $ab=2$ and $a+b=1$
$\implies a,b=?$
Use $x^2+y^2=(x+y)^2-2xy$ twice to reach at $$a^4+b^4=1$$
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