jmc

We know that $\lfloor x\rfloor \le x<\lfloor x\rfloor +1$. This implies that $\lfloor x{\rfloor }^2\le x\cdot \lfloor x\rfloor <{\left(\lfloor x\rfloor +1\right)}^2$ for all values of $x$. In particular since $x\cdot \lfloor x\rfloor =70$ and $8^2<70<9^2$, we can conclude that $8<x<9⟹\lfloor x\rfloor =8$. From there, all we have to do is divide to get that $x=\frac{70}{8}=\boxed{8.75}$.