Question

x^2-1/x=4 find value x^6-1/x^6=?​

Answer 1

Given,

${x}^{2} - \frac{1}{ {x}^{2} } = 4 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \cdots(1)$

Then,

Cubing both sides of Equation 1, we get,

${ \bigg( {x}^{2} - \frac{1}{ {x}^{2} } \bigg )}^{3} = {(4)}^{3} \\ \\ {x}^{6} - \frac{1}{ {x}^{6} } - 3 \bigg( {x}^{2} \bigg) \bigg( \frac{1}{ {x}^{2} } \bigg) \bigg\{ {x}^{2} - \frac{1}{ {x}^{2} } \bigg\} = 64 \\ \\ {x}^{6} - \frac{1}{ {x}^{6} } - 3(4) = 64 \: \: \: \: \: \: \: \: \: \: \: \: \big\{ \because {x}^{2} - \frac{1}{ {x}^{2} } = 4 \big\}\\ \\ {x}^{6} - \frac{1}{ {x}^{6} } - 12 = 64 \\ \\ {x}^{6} - \frac{1}{ {x}^{6} } = 64 + 12 \\ \\ \huge{x}^{6} - \frac{1}{ {x}^{6} } = \pink{ 78}$