### Instruction: Simplify $\displaystyle\frac{24t^3}{15t^4}\cdot \frac{5t^8}{3t^6}$.### Response: I notice that this is a product of two fractions, so I can use the rule that $\displaystyle\frac{a}{b}\cdot \frac{c}{d} = \frac{ac}{bd}$.
I also notice that the fractions have powers of $t$ in the numerator and denominator, so I can use the rule that $\displaystyle\frac{t^m}{t^n} = t^{m-n}$.
Applying these rules, I get $\displaystyle\frac{24t^3}{15t^4}\cdot \frac{5t^8}{3t^6} = \frac{24\cdot 5}{15\cdot 3} \cdot \frac{t^3\cdot t^8}{t^4\cdot t^6} = \frac{120}{45} \cdot \frac{t^{11}}{t^{10}}$.
To simplify the coefficient, I divide 120 by 45 and get 8/3.
To simplify the power of $t$, I subtract the exponents and get $t^1$ or just $t$.
Therefore, the final answer is $\displaystyle\frac{8}{3}t$.