Question

If x+1/x=root3 find x^3 +1/x^3

Answer 1

        $\text{It is given that }x + \dfrac1{x} = \sqrt3$

        $\text{cubing both sides}$

$\implies \: \text{\huge{(}}x + \dfrac1{x}\text{\huge{)}}^3 = (\sqrt3)^3$

$\implies \: \text{\huge{(}}x + \dfrac1{x}\text{\huge{)}}^3 = 9\sqrt3$

        $\text{We know that }(a+b)^3 = a^3 + b^3 + 3ab(a+b)$

$\implies \: x^3 + \dfrac{1}{x^3} + 3(x)\text{\huge{(}}\dfrac1{x^}\text{\huge{)}}\text{\huge{(}}x + \dfrac1{x}\text{\huge{)}} = 9\sqrt3$

$\implies \: x^3 + \dfrac{1}{x^3} + 3\text{\huge{(}}x + \dfrac1{x}\text{\huge{)}} = 9\sqrt3$

        $\text{Given that }x + \dfrac1{x} = \sqrt3$

$\implies \: x^3 + \dfrac{1}{x^3} + 3\sqrt3 = 9\sqrt3$

$\implies \: \boxed{x^3 + \dfrac{1}{x^3}= 6\sqrt3}$