The expression you're working with is:

$(-3x^2y^3z^2)(-3xy^3z^4)^3$

Let's break it down:

Step 1: Apply the power rule to $(-3xy^3z^4)^3$

When you raise a product to a power, you apply the exponent to each term inside the parentheses:

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

Simplifying each part:

$(-3)^3 = -27, \quad (x)^3 = x^3, \quad (y^3)^3 = y^9, \quad (z^4)^3 = z^{12}$

So:

$(-3xy^3z^4)^3 = -27x^3y^9z^{12}$

Step 2: Multiply the two parts

Now, we multiply $(-3x^2y^3z^2)$ by $-27x^3y^9z^{12}$:

$(-3x^2y^3z^2) \cdot (-27x^3y^9z^{12})$

We multiply the constants, $x$-terms, $y$-terms, and $z$-terms separately:

$\text{Constant: } (-3) \cdot (-27) = 81$ $x\text{-terms: } x^2 \cdot x^3 = x^{2+3} = x^5$ $y\text{-terms: } y^3 \cdot y^9 = y^{3+9} = y^{12}$ $z\text{-terms: } z^2 \cdot z^{12} = z^{2+12} = z^{14}$

Step 3: Final expression

Thus, the simplified expression is:

$81x^5y^{12}z^{14}$

Would you like more details or have any questions about the steps?

Here are some related questions you can explore:

1. How do you handle negative exponents in multiplication?
2. What is the rule for multiplying exponents with the same base?
3. How do you simplify expressions with multiple variables raised to powers?
4. What happens when you multiply a number raised to an exponent by another number raised to an exponent?
5. How do you simplify more complex expressions with multiple terms?

Tip: Always apply the exponent rules carefully, especially when dealing with negative numbers and multiple variables.