If $ r + \frac{1}{r} =11. $ then the value of $\frac{r^3+\frac{1}{r}}{r^2-r+1}$ is _________.

$\frac{119}{10}$

If $ r + \frac{1}{r} =11$

Then the value of $\frac{r^3+\frac{1}{r}}{r^2-r+1}$

Now, $ r^2 + \frac{1}{r^2} =11^2 - 2$ = 119

Divide $\frac{r^3+\frac{1}{r}}{r^2-r+1}$ by r on numerator and denominator then we get,

= $\frac{r^2+\frac{1}{r^2}}{r+\frac{1}{r}-1}$

= $\frac{119}{11-1}$

= $\frac{119}{10}$