Certainly! Let's tackle part (a) and part (b) step-by-step.
### Part (a): Simplify [tex]\(\frac{12 t^2}{v} \div \frac{2 t^5}{v^3}\)[/tex]
To simplify [tex]\(\frac{12 t^2}{v} \div \frac{2 t^5}{v^3}\)[/tex], follow these steps:
1. Rewrite the Division as Multiplication by the Reciprocal:
[tex]\[
\frac{12 t^2}{v} \div \frac{2 t^5}{v^3} = \frac{12 t^2}{v} \times \frac{v^3}{2 t^5}
\][/tex]
2. Combine the Fractions:
[tex]\[
\frac{12 t^2 \cdot v^3}{v \cdot 2 t^5}
\][/tex]
3. Simplify the Coefficients:
[tex]\[
\frac{12 \cdot v^3}{2 \cdot v} = \frac{12}{2} \cdot \frac{v^3}{v}
\][/tex]
[tex]\[
\frac{12}{2} = 6
\][/tex]
4. Simplify the Variable [tex]\(v\)[/tex]:
[tex]\[
\frac{v^3}{v} = v^{3-1} = v^2
\][/tex]
5. Combine [tex]\( t \)[/tex]-Terms:
[tex]\[
\frac{t^2}{t^5} = t^{2-5} = t^{-3}
\][/tex]
6. Combine All Simplified Parts:
Putting it all together, we get:
[tex]\[
6 \cdot v^2 \cdot t^{-3} = 6 v^2 t^{-3}
\][/tex]
Thus, the simplified form of [tex]\(\frac{12 t^2}{v} \div \frac{2 t^5}{v^3}\)[/tex] is:
[tex]\[
6 v^2 t^{-3}
\][/tex]
### Part (b): Solve
Since part (b) doesn't provide specific details about what needs to be solved, I'll need more information to proceed accurately. If you have a specific equation or additional context for part (b), please provide it, and I'll be glad to help you solve it. For now, let's conclude part (a) as:
[tex]\[
\boxed{6 v^2 t^{-3}}
\][/tex]
