Let's simplify the given expression:

$(4x^3 y^0 z^5)(5x y^{-3} z^9)^{-2}$

Step 1: Simplify the First Term

Since $y^0 = 1$, the first term simplifies to:

$4x^3 z^5$

Step 2: Apply the Exponent to the Second Term

The exponent $-2$ applies to the entire term $5x y^{-3} z^9$, so we distribute it:

$(5x y^{-3} z^9)^{-2} = 5^{-2} x^{-2} y^{6} z^{-18}$

Since $5^{-2} = \frac{1}{25}$, we rewrite it as:

$\frac{x^{-2} y^6 z^{-18}}{25}$

Step 3: Multiply the Terms

$(4x^3 z^5) \times \left(\frac{x^{-2} y^6 z^{-18}}{25} \right)$

Multiply the coefficients:

$\frac{4}{25}$

For $x$:

$x^{3 + (-2)} = x^1 = x$

For $y$:

$y^6$

For $z$:

$z^{5 + (-18)} = z^{-13} = \frac{1}{z^{13}}$

Final Answer:

$\frac{4x y^6}{25z^{13}}$

Would you like further clarification on any step?

Related Questions:

1. How do you simplify expressions with negative exponents?
2. What are the exponent rules for multiplying and dividing powers?
3. How does a zero exponent affect a term?
4. How do you handle fractions in exponentiation?
5. What happens when you raise a product to a negative power?

Tip:

Always simplify individual terms first before applying exponent rules to avoid mistakes!