To solve the expression
\[ \frac{\frac{3t^5}{3t^6}}{\frac{(3t)^2}{-15}}, \]
we'll first simplify each part separately.
- Simplify the numerator:
\[ \frac{3t^5}{3t^6} = \frac{3}{3} \cdot \frac{t^5}{t^6} = 1 \cdot t^{5-6} = t^{-1} = \frac{1}{t} \]
- Simplify the denominator:
First, calculate \((3t)^2\):
\[ (3t)^2 = 9t^2. \]
Now, substitute this back into the expression for the denominator:
\[ \frac{(3t)^2}{-15} = \frac{9t^2}{-15} = -\frac{9t^2}{15}. \]
Next, simplify \(-\frac{9t^2}{15}\):
\[ -\frac{9t^2}{15} = -\frac{3t^2}{5} \quad (\text{dividing the numerator and denominator by 3}). \]
- Putting it all together:
Now we substitute the simplified numerator and denominator back into the original expression:
\[ \frac{\frac{1}{t}}{-\frac{3t^2}{5}} = \frac{1}{t} \cdot \frac{5}{-3t^2} = \frac{5}{-3t^3} = -\frac{5}{3t^3}. \]
Thus, the final answer in lowest terms is:
\[ -\frac{5}{3t^3}. \]