For a natural number k let Tk = xk + 1/xk. Note that T4T2 = T2+T6 and T8T2 = T10+T6. Therefore T2 (T8 - T4 + 1 ) = T10. Since T2k = Tk2 + 2, it follows that T8, T10 are rational numbers and hence T2, T6 are also rational numbers. Since T5T1 = T4+T6 it follows that T1 is a rational number.