Question

if x-1/x=√5,find the values of
1) x^2+1/x^2
2)x+1/x
3)x^3+1/x^3​

Answer 1

$x - \frac{1}{x} = \sqrt{5} \: \: \: (given) \\ \\ 1) \: {x}^{2} + \frac{1}{ {x}^{2} } = (x - \frac{1}{x} )^{2} + 2 \\ = {( \sqrt{5}) }^{2} + 2 \\ = 5 + 2 \\ = 7$

$2) \: {(x + \frac{1}{x} )}^{2} = {x}^{2} + \frac{1}{ {x}^{2} } + 2 \\ x + \frac{1}{x} = \sqrt{7 + 2} \\ x + \frac{1}{x} = \sqrt{9} \\ x + \frac{1}{x} = 3$

${x}^{3} + \frac{1}{ {x}^{3} } = (x + \frac{1}{x} )( {x}^{2} + \frac{1}{ {x}^{2} } - 1) \\ = 3(7 - 1) \\ = 3 \times 6 \\ = 18$