If x+(1)/(x)=√(11), find x^(3)+(1)/(x^(3))

Question

If x+
`(1)/(x)`=
`√(11)`, find
`x^(3)`+
`(1)/(x^(3))`

Answer 1

Given that ,
`x +(1)/(x)=√(11)`

And we are interested in finding out the value of
`x^3+(1)/(x^3)`

Taking the given equation,

`\longrightarrow x +(1)/(x)=√(11)`

cube both the sides ,

`\longrightarrow \left( x +(1)/(x)\right)^3=(√(11))^3\\`

simplify using identity,

`\longrightarrow x^3+(1)/(x^3)+3(x)\bigg((1)/(x)\bigg)\bigg( x +(1)/(x)\bigg)=11√(11)\\`

`\longrightarrow x^3+(1)/(x^3)+3√(11)= 11√(11)\\`

`\longrightarrow x^3+(1)/(x^3)=11√(11)-3√(11)\\`

`\longrightarrow \boxed{x^3+(1)/(x^3)= 8√(11) }`

And we are done!