Question

$x + \frac{1}{x} = 3 \\ \\ x {8} + \frac{1}{{x}^{8} } =?$

Answer 1

$x + \frac{1}{x} = 3 \\ {x}^{2} + \frac{1}{ {x}^{2} } = {(x + \frac{1}{x} )}^{2} - 2 = {3}^{2} - 2 = 7 \\ {x}^{4} + \frac{1}{ {x}^{4} } = {( {x}^{2} + \frac{1}{ {x}^{2} } )}^{2} - 2 = {7}^{2} - 2 = 47 \\ {x}^{8} + \frac{1}{ {x}^{8} } = {( {x}^{4} + \frac{1}{ {x}^{4} } )}^{2} - 2 = {47}^{2} - 2 = 2207$