If $x^{2}-9 x+1=0$, what is the value of $\left(x^{3}+\frac{1}{x^{3}}\right) ?$

$x^{2}-9 x+1=0$

$\Rightarrow x^{2}+1=9 x$

$\Rightarrow \frac{x^{2}+1}{x}=9$

$\Rightarrow x+\frac{1}{x}=9$

On cubing both sides,

$\left(x+\frac{1}{x}\right)^{3}=9^{3}=729$

$\Rightarrow x^{3}+\frac{1}{x^{3}}+3\left(x+\frac{1}{x}\right)=729$

$\Rightarrow x^{3}+\frac{1}{x^{3}}+3 \times 9=729$

$\Rightarrow x^{3}+\frac{1}{x^{3}}=729-27=702$