Question

Divide. Write the answer in lowest terms.StartFraction 3 t Superscript 5 Over 3 t Superscript 6 EndFraction divided by StartFraction left parenthesis 3 t right parenthesis squared Over negative 15 EndFraction

Answer 1

To solve the expression

\[
\frac{\frac{3t^5}{3t^6}}{\frac{(3t)^2}{-15}},
\]

we'll first simplify each part separately.

1. **Simplify the numerator**:

\[
\frac{3t^5}{3t^6} = \frac{3}{3} \cdot \frac{t^5}{t^6} = 1 \cdot t^{5-6} = t^{-1} = \frac{1}{t}
\]

2. **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 \(-\frac{9t^2}{15}\):

\[
-\frac{9t^2}{15} = -\frac{3t^2}{5} \quad (\text{dividing the numerator and denominator by 3}).
\]

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:

\[
-\frac{5}{3t^3}.
\]