Question

If x+1/x=6 find x2+1/x2,x3+1/x3,x4+1/x4

Answer 1

$x + \frac{1}{x} = 6 \\ {( x + \frac{1}{x} ) }^{2} = {(6)}^{2} \\ {x}^{2} + \frac{1}{ {x}^{2} } + 2x( \frac{1}{x} ) = 36 \\ {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 36 \\ {x}^{2} + \frac{1}{ {x}^{2} } = 34 \\ \\ {( {x}^{2} + \frac{1}{ {x}^{2} }) }^{2} = {(34)}^{2} \\ {x}^{4} + \frac{1}{ {x}^{4} } + 2 {x}^{2} ( \frac{1}{ {x}^{2} } ) = 1156 \\ {x}^{4} + \frac{1}{ {x}^{4} } + 2 = 1156 \\ {x}^{4} + \frac{1}{ {x}^{4} } = 1154 \\ \\ x + \frac{1}{x} = 6 \\ {(x + \frac{1}{x} )}^{3} = {(6)}^{3} \\ {x}^{3} + \frac{1}{ {x}^{3} } + 3x( \frac{1}{x} )(x + \frac{1}{x} ) = 216 \\ {x}^{3} + \frac{1}{{x}^{3} } + 3(6) = 216 \\ {x}^{3} + \frac{1}{ {x}^{3} } + 18 = 216 \\ {x}^{3} + \frac{1}{ {x}^{3} } = 198$