+130511 x + x^-1 = √2 multiply through by x
x^2 + 1 = √2x
x^2 - √2x = - 1 complete the square on x
x^2 - √2/x + 1/2 = -1 + 1/2
[ x - (1/√2)]^2 = -1/2 take both roots
x - (1/√2 ) = ±i/ √2
x = ( 1 + i) x = ( 1 - i )
______ or ________
√2 √2
x ^20 = [ ( 1 + i)) ] ^20
____________
( √2)^20
Note that ( 1 + i)^2 = 2i .....so we have
[ (1 + i)^2 ] ^10 [ 2i ] ^10
x^20 = _______________ = ________ = i^10 = -1
[( √2)^2]^10 2^10
The same result is obtained for the other solution for x raised to the 20th power
