Let's simplify the expression step by step:

Given:
$(-2xy)^4 \times (-3x^2)^3$

Step 1: Simplify each term separately

Term 1: $(-2xy)^4$

Use the rule of exponents $(ab)^n = a^n \cdot b^n$:
$(-2xy)^4 = (-2)^4 \cdot x^4 \cdot y^4$
Since $(-2)^4 = 16$, the first term becomes: $(-2xy)^4 = 16x^4y^4$

Term 2: $(-3x^2)^3$

Again, use the rule $(ab)^n = a^n \cdot b^n$:
$(-3x^2)^3 = (-3)^3 \cdot (x^2)^3$
Since $(-3)^3 = -27$ and $(x^2)^3 = x^{2 \cdot 3} = x^6$, the second term becomes: $(-3x^2)^3 = -27x^6$

Step 2: Multiply the two terms

Now multiply the results from Step 1:
$(16x^4y^4) \cdot (-27x^6)$

Combine coefficients: $16 \cdot (-27) = -432$

Combine $x$-terms using the rule $x^m \cdot x^n = x^{m+n}$:
$x^4 \cdot x^6 = x^{4+6} = x^{10}$

The $y^4$ term remains as it is because there are no other $y$-terms:
$y^4$

Final Expression:

$-432x^{10}y^4$

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Let me know if you need further clarification or have any questions!

Related Questions:

1. What is the general rule for simplifying powers of products, like $(ab)^n$?
2. How do negative exponents affect the results of expressions?
3. What happens if one of the bases in the problem is zero?
4. Can this process be applied to fractional exponents? How?
5. What are some real-life applications of exponential expressions?

Tip:

Always break down powers step by step and apply the exponent rules systematically to avoid mistakes.