.
Consider complex number
z = 
. (1)
It is the root of the degree 7 of 1 in complex domain, so
= 1 = 
. (2)
Apply the binomial decomposition to get
1 = 
+
+
+
+
+
+ 
+
.
In the last equation extract the imaginary part, which is equal to zero:
0 = + + + .
Replace the degrees of "i" by i with the corresponding signs and then divide both sides by 
. You will get
0 = 
- 
+ 
- 
.
Divide by 
both sides. You will get
- 
+ 
- 7 = 0. (3)
Thus you see that 
is the root of the polynomial equation
- 
+ 
- 7 = 0. (4)
If you do the same starting from z = 
instead of z = , you will get by the same way that
is the root of the same polynomial equation.
Similarly, If you do the same starting from z = 
instead of z = , you will get by the same way that
is the root of the same polynomial equation.
. . . and so on . . .
Thus, the six numbers , , , . . . , 
all are the roots of the equation (4).
Then, according to Vieta's theorem, the product of the roots is equal to the constant term:
... . . . . = -7.
Now take into account that = -, = -
and = -
.
Based on it, you get
.
.
= 7,
which implies
.. = 
.
It is exactly what has to be proved.
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* * * SOLVED. * * *
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