Let's simplify the expression $(-3x^3y^4)^4$.

We will apply the power of a product rule, which states that $(ab)^n = a^n b^n$. Here we have:

$(-3x^3y^4)^4 = (-3)^4 \cdot (x^3)^4 \cdot (y^4)^4$

Now, let's break it down:

1. $(-3)^4 = 81$
2. $(x^3)^4 = x^{12}$ (applying the power of a power rule: $(a^m)^n = a^{m \cdot n}$)
3. $(y^4)^4 = y^{16}$ (again, applying the same rule)

Putting it all together:

$(-3x^3y^4)^4 = 81x^{12}y^{16}$

Thus, the fully simplified expression is:

$81x^{12}y^{16}$

Do you have any questions or would you like more details on the steps?

Here are some related questions you might find interesting:

1. What is the rule for multiplying powers with the same base?
2. How do you simplify negative exponents?
3. What is the difference between $(x^m)^n$ and $x^{m \cdot n}$?
4. How would you simplify expressions involving multiple terms raised to a power?
5. What happens when you raise a term with an exponent of 0 to a power?

Tip: When simplifying expressions with exponents, always break the expression into separate parts and simplify each part individually.