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The given expression appears to be a fraction containing products of variables raised to powers. Let me break it down step by step:

\[
\left( \frac{(3x^{-4}y^4z^2)(x^3y^{-2}z)}{-2x^3y^4z^3} \right) - 3
\]

### Step 1: Simplify the numerator
The numerator is:
\[
(3x^{-4}y^4z^2)(x^3y^{-2}z)
\]
Apply the laws of exponents when multiplying powers of the same base:
- \( x^{-4} \cdot x^3 = x^{(-4+3)} = x^{-1} \)
- \( y^4 \cdot y^{-2} = y^{(4-2)} = y^2 \)
- \( z^2 \cdot z = z^{(2+1)} = z^3 \)

Thus, the numerator simplifies to:
\[
3x^{-1}y^2z^3
\]

### Step 2: Simplify the denominator
The denominator is:
\[
-2x^3y^4z^3
\]

### Step 3: Combine the fraction
Now combine the numerator and the denominator:
\[
\frac{3x^{-1}y^2z^3}{-2x^3y^4z^3}
\]

Apply the laws of exponents for division:
- \( \frac{x^{-1}}{x^3} = x^{-1-3} = x^{-4} \)
- \( \frac{y^2}{y^4} = y^{2-4} = y^{-2} \)
- \( \frac{z^3}{z^3} = z^{3-3} = z^0 = 1 \)

Thus, the entire fraction simplifies to:
\[
\frac{3}{-2} x^{-4} y^{-2}
\]
This is equivalent to:
\[
-\frac{3}{2} \cdot \frac{1}{x^4 y^2}
\]

### Step 4: Subtract 3
Now, subtract 3 from the simplified expression:
\[
-\frac{3}{2x^4 y^2} - 3
\]

This is the fully simplified form of the given expression.

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Would you like more details or have any questions?

Here are some related questions you might find useful:
1. How do you apply the laws of exponents when multiplying variables?
2. Can negative exponents be expressed in fractional form?
3. What happens when exponents cancel out, like \( z^3 / z^3 \)?
4. How can you simplify complex fractions with variables?
5. How do you combine terms with different bases in algebraic expressions?

**Tip:** When simplifying algebraic expressions, always deal with exponents carefully, applying the rules of addition and subtraction based on the operations involved (multiplication or division).

((3x^(-4)y^4z^2)(x^3y^(-2)z))/(-2x^3y^4z^3) - 3

Step-by-step breakdown of how to simplify the algebraic expression ((3x^(-4)y^4z^2)(x^3y^(-2)z))/(-2x^3y^4z^3) - 3 using laws of exponents. This problem covers key algebraic concepts involving exponents and fractions, suitable for students in Grades 9-12.

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