Let's simplify the expression:

$\frac{4x^0 y^{-2} z^3}{4x}.$

Step 1: Apply the exponent rules

• $x^0 = 1$, because any number raised to the power of 0 is 1.
• The coefficient $4$ on both the numerator and the denominator cancels out.

Thus, the expression becomes:

$\frac{1 \cdot y^{-2} \cdot z^3}{x}.$

Step 2: Simplify further

Now, let's rewrite it without the unnecessary coefficients:

$\frac{y^{-2} z^3}{x}.$

Step 3: Handle the negative exponents

Since $y^{-2}$ is in the numerator, it can be rewritten as $\frac{1}{y^2}$ in the denominator. Therefore, the expression becomes:

$\frac{z^3}{x y^2}.$

This is the simplified form of the given expression.

Would you like a further explanation of any steps or need more details?

Here are some related questions you might find helpful:

1. How do negative exponents affect the placement of variables in a fraction?
2. What are the properties of exponents when dividing terms with the same base?
3. How do you simplify expressions with exponents that are fractions?
4. What happens when an exponent is zero for a term in an equation?
5. How can you multiply or divide terms with the same base in an expression?

Tip: When simplifying expressions with exponents, always remember to apply the exponent rules carefully, especially when negative exponents are involved.