If x+(1)/(x)=√(11)\\, find the value of x^(3)+(1)/(x^(3))

Question

if x+
`(1)/(x)`=
`√(11)\\`, find the value of
`x^(3)`+
`(1)/(x^(3))`

Answer 1

`8 √(11)`

Answer 2

`8√(11)`

Explanation:

Given:

`x+(1)/(x)=√(11)`

Sum of two squares

`\boxed{a^2+b^2=(a+b)^2-2ab}`

`\textsf{Let}\; a = x`

`\textsf{Let}\; b = (1)/(x)`

Therefore:

`\begin{aligned}x^2+(1)/(x^2)&=\left(x+(1)/(x)\right)^2-2x\left((1)/(x)\right)\\\\&=\left(√(11)\right)^2-2\\\\&=11-2\\\\&=9\end{aligned}`

Sum of two cubes

`\boxed{a^3+b^3=(a+b)(a^2-ab+b^2)}`

`\textsf{Let}\; a = x`

`\textsf{Let}\; b = (1)/(x)`

Therefore:

`\begin{aligned}x^3+(1)/(x^3)&=\left(x+(1)/(x)\right)\left(x^2-x\left((1)/(x)\right)+\left((1)/(x)\right)^2\right)\\\\ &=\left(x+(1)/(x)\right)\left(x^2-1+(1)/(x^2)\right)\\\\&=\left(x+(1)/(x)\right)\left(x^2+(1)/(x^2)-1\right)\\\\&=√(11)\left(9-1\right)\\\\&=8√(11)\end{aligned}`