1. **Problem Statement:** We need to find the values of powers $n$ and $m$ such that the acceleration $a_s$ of a particle moving with uniform speed $v$ in a circle of radius $r$ is proportional to $r^n$ and $v^m$. Then, write the simplest form of the equation for $a_s$. 2. **Known facts:** For uniform circular motion, the acceleration (centripetal acceleration) is directed towards the center and depends on the speed and radius. 3. **Formula:** The centripetal acceleration is given by $$a_s = \frac{v^2}{r}$$ 4. **Expressing in terms of powers:** We can rewrite this as $$a_s = v^2 \cdot r^{-1}$$ 5. **Comparing powers:** From the expression, the power of $v$ is $m=2$ and the power of $r$ is $n=-1$. 6. **Final equation:** Therefore, the simplest form of the equation is $$a_s = k v^2 r^{-1}$$ where $k$ is the proportionality constant (usually $k=1$ in standard units). **Summary:** - $n = -1$ - $m = 2$ - Equation: $$a_s = k v^2 r^{-1}$$