Simplify. Express your answer using positive exponents. \( \frac{4 r^{3} s^{6} t}{\left(2 r^{5} s^{3} t^{7}\right)\left(r^{2} s t\right)} \)

To simplify the expression \( \frac{4 r^{3} s^{6} t}{\left(2 r^{5} s^{3} t^{7}\right)\left(r^{2} s t\right)} \), we start by simplifying the denominator. First, multiply the two expressions in the denominator: \[ \left(2 r^{5} s^{3} t^{7}\right)\left(r^{2} s t\right) = 2 r^{5+2} s^{3+1} t^{7+1} = 2 r^{7} s^{4} t^{8} \] Now our expression looks like this: \[ \frac{4 r^{3} s^{6} t}{2 r^{7} s^{4} t^{8}} \] Next, simplify the coefficients and the variables separately: 1. The coefficients: \( \frac{4}{2} = 2 \) 2. For \( r \): \[ \frac{r^{3}}{r^{7}} = r^{3-7} = r^{-4} \] 3. For \( s \): \[ \frac{s^{6}}{s^{4}} = s^{6-4} = s^{2} \] 4. For \( t \): \[ \frac{t^{1}}{t^{8}} = t^{1-8} = t^{-7} \] Putting it all together, we have: \[ 2 r^{-4} s^{2} t^{-7} \] Finally, we express the answer with positive exponents: \[ \frac{2 s^{2}}{r^{4} t^{7}} \] So the simplified expression is: \[ \frac{2 s^{2}}{r^{4} t^{7}} \]