Question

if x^2+1/x^=14, using only the positive value of x+1/x then find x^3+1/x^3

Answer 1

$\textbf{ Hello Mate! }$

${x}^{2} + \frac{1}{ {x}^{2} } = 14 \: (given) \\ {(x + \frac{1}{x}) }^{2} = {x }^{2} + \frac{1}{ {x}^{2} } + 2(x)( \frac{1}{x} ) \\ {(x + \frac{1}{x}) }^{2} = 14 + 2 \\ x + \frac{1}{ x } = \sqrt{16 } = 4 \\ \\ {a}^{3} + {b}^{3} = {(a + b)}^{3} - 3ab(a + b) \\ {x}^{3} + \frac{1}{ {x}^{3} } = {(x + \frac{1}{x} )}^{3} - 3(x)( \frac{1}{x} )(x + \frac{1}{x} ) \\ = {(14)}^{3} - 3(14) \\ = 2744 - 42 \\ = 2702$

$\boxed{ \textsf{ \red{ =>\:2702 }}}$

Have great future ahead!