If $x^2+x+1=0 $, then find the value of $(x^3+1/x^3)^3$? 
If $x^2+x+1=0 $, then find the value of $(x^3+1/x^3)^3$

This thing doesn't make sense how should I use first identity to find the second one.

The response of @TonyK is perfect and exemplary. Since some seem not to follow it, let me reinterpret it. By using Euclidean Division on $X^2+X+1$ and $X^3$, you get the identity $X^3=(X^2+X+1)(X-1) + 1$. This identity is still valid when any number is substituted for $X$. In particular, when you substitute that $x$ for which $x^2+x+1=0$. So $x^3=1$, and $1/x^3=1$.
