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.

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).