To simplify the expression:

$\frac{3y^4z^{-3}}{x^0y^{-4}z^3}$

Step-by-step solution:

1. Handle $x^0$: Any non-zero number raised to the power of 0 is 1. So, $x^0 = 1$. The denominator becomes:

   $y^{-4}z^3$

   The expression is now:

   $\frac{3y^4z^{-3}}{y^{-4}z^3}$

2. Simplify the $y$-terms: Use the rule of exponents: $\frac{a^m}{a^n} = a^{m-n}$. For $y^4$ and $y^{-4}$:

   $y^{4 - (-4)} = y^{4 + 4} = y^8$

3. Simplify the $z$-terms: Similarly, for $z^{-3}$ and $z^3$:

   $z^{-3 - 3} = z^{-6}$

4. Combine the results: The expression becomes:

   $3y^8z^{-6}$

5. Write with positive exponents: To express $z^{-6}$ with a positive exponent, rewrite it as $\frac{1}{z^6}$. The final result is:

   $\frac{3y^8}{z^6}$

Final Answer:

$\boxed{\frac{3y^8}{z^6}}$

Would you like further details or have any questions?
Here are 5 related questions to explore:

1. How do you simplify expressions with fractional exponents?
2. What is the general rule for negative exponents?
3. How do you handle expressions involving multiple variables in both the numerator and denominator?
4. Why does $x^0 = 1$, and what exceptions might there be?
5. How do you simplify expressions with exponents involving addition or subtraction?

Tip: Always combine like terms and simplify step-by-step to avoid errors.