jmc

We realize that $a^3+b^3$ is the sum of two cubes and thus can be expressed as $(a+b)(a^2-ab+b^2)$. From this, we have $$\begin{array}{rl}{a}^3+b^3& =(a+b)(a^2-ab+b^2)\\ & =(a+b)((a^2+2ab+b^2)-3ab)\\ & =(a+b)((a+b{)}^2-3ab)\end{array}$$Now, since $a+b=10$ and $ab=17$, we have $$a^3+b^3=(a+b)((a+b{)}^2-3ab)=10\cdot (10^2-3\cdot 17)=10\cdot 49=\boxed{490}.$$