Find the positive real number $x$ such that $\lfloor x \rfloor \cdot x = 70$. Express $x$ as a decimal.
Knowing that $\lfloor x \rfloor \leq x < \lfloor x \rfloor + 1,$ we see that $\lfloor x \rfloor$ must be $8,$ since $8 \cdot 8 \leq 70 < 9 \cdot 9.$

Now we see that $\lfloor x \rfloor \cdot x = 8x = 70,$ so $x = \frac{70}{8} = \boxed{8.75}.$