Question

Find r^3+1/r^3 if r+1/r=sqrt2

Answer 1

Recall that

`(a+b)^3=a^3+3a^2b+3ab^2+b^3`

which means

`\left(r+\frac1r\right)^3=r^3+3r^2\left(\frac1r\right)+3r\left(\frac1r\right)^2+\left(\frac1r\right)^3=r^3+3r+\frac3r+\frac1{r^3}`

Given that
`r+\frac1r=\sqrt2`, we have

`(\sqrt2)^3=r^3+3\sqrt2+\frac1{r^3}`

`\implies r^3+\frac1{r^3}=2\sqrt2-3\sqrt2=\boxed{-\sqrt2}`