Question

If x^2+1/x^2=7, find the value of x^3+1/x^3

Answer 1

$\rm{given \: {x}^{2} + \frac{1}{ {x}^{2} } = 7}$

$\rm{{(x + \frac{1}{x} )}^{2} = {x}^{2} + \frac{1}{ {x}^{2} } + 2}$

$\implies \rm{{(x + \frac{1}{x} )}^{2} = 7 + 2}$

$\implies\rm{{(x + \frac{1}{x} )}^{2} = 9}$

$\implies\rm{x + \frac{1}{x} = \sqrt{9} }$

$\implies\rm{x + \frac{1}{x} = 3}$

$\sf{cubing \: on \: both \: sides}$

$\rm{(x + \frac{1}{x})}^{3} = {3}^{3}$

$\implies\rm{ {x}^{3} + \frac{1}{ {x}^{3} } + 3(x + \frac{1}{x} ) } = 27$

$\implies\rm{ {x}^{3} + \frac{1}{ {x}^{3} } + 3(3) } = 27$

$\implies\rm{ {x}^{3} + \frac{1}{ {x}^{3} } + 9} = 27$

$\implies\rm{ {x}^{3} + \frac{1}{ {x}^{3} } } = 27 - 9$

$\fbox{\rm{ {x}^{3} + \frac{1}{ {x}^{3} } } = 18}$