If $$x + \frac{1}{x} = 2$$   and x is real, then the value of $${x^{17}}{\text{ + }}\frac{1}{{{x^{19}}}}\,{\text{is?}}$$

$$\eqalign{ & x + \frac{1}{x} = 2 \cr & \left( {{\text{Assume }}x = 1{\text{, so, }}1 + 1 = 2} \right) \cr & {x^{17}}{\text{ + }}\frac{1}{{{x^{19}}}} \cr & = {\left( 1 \right)^{17}} + \frac{1}{{{{\left( 1 \right)}^{19}}}} \cr & = 1 + 1 \cr & = 2 \cr} $$