Let's solve the expression given in the image:

$\frac{3^{-5} \cdot (3^3)^{3.81} \cdot 81}{3^6}$

Step 1: Simplify the Exponents

1. Simplify $(3^3)^{3.81}$:

   $(3^3)^{3.81} = 3^{3 \cdot 3.81} = 3^{11.43}$

2. Simplify $81$ as a power of 3:

   $81 = 3^4$

   So the expression now becomes:

   $\frac{3^{-5} \cdot 3^{11.43} \cdot 3^4}{3^6}$

Step 2: Combine the Exponents

Use the property $a^m \cdot a^n = a^{m+n}$:

$\frac{3^{-5 + 11.43 + 4}}{3^6} = \frac{3^{10.43}}{3^6}$

Step 3: Simplify the Division

Use the property $\frac{a^m}{a^n} = a^{m-n}$:

$3^{10.43 - 6} = 3^{4.43}$

Step 4: Compare to Answer Choices

The given options are in the form of powers of 3. Since $3^{4.43}$ is approximately $3^4$, and there is no direct option that perfectly matches this, but if we round it down, the closest simpler option among the choices would be $3^3$. However, the calculation suggests that the closest match should be closer to $3^4$.

Given the exact context, the provided calculation suggests rounding towards $3^3$ might be intentional as an approximate answer.

Conclusion

The answer selected in the image is $3^3$, but based on the computation, $3^{4.43}$ would round to $3^4$.

Would you like more details or have any further questions?

Related questions:

1. How does rounding affect exponentiation results?
2. What are other ways to express non-integer exponents?
3. How do powers of 3 compare in magnitude?
4. Why is it important to simplify exponents before solving?
5. Can you derive the steps to approximate exponents without a calculator?

Tip: When working with exponents, always look for ways to simplify the terms first; this often makes the problem much easier to solve.