1. **State the problem:** Simplify the expression $$\frac{(xyz^2)^{-1}}{x^2y^{-1}z^3} \div \left(\frac{x^3y^2z^0}{x^4y^2z^{-3}}\right)^{-1} \times \frac{xy^2z^4}{x^{-3}y^2z^0}$$ 2. **Recall the rules:** - Negative exponents: $a^{-n} = \frac{1}{a^n}$ - Division of powers: $\frac{a^m}{a^n} = a^{m-n}$ - Multiplication of powers: $a^m \times a^n = a^{m+n}$ - Power of a power: $(a^m)^n = a^{mn}$ - $z^0 = 1$ 3. **Simplify each part:** - First fraction numerator: $(xyz^2)^{-1} = x^{-1}y^{-1}z^{-2}$ - First fraction denominator: $x^2 y^{-1} z^3$ So first fraction: $$\frac{x^{-1} y^{-1} z^{-2}}{x^2 y^{-1} z^3} = x^{-1-2} y^{-1-(-1)} z^{-2-3} = x^{-3} y^{0} z^{-5} = \frac{x^{-3} z^{-5}}{1}$$ 4. **Simplify second fraction inside parentheses:** $$\frac{x^3 y^2 z^0}{x^4 y^2 z^{-3}} = x^{3-4} y^{2-2} z^{0-(-3)} = x^{-1} y^{0} z^{3} = x^{-1} z^{3}$$ 5. **Apply the negative exponent outside parentheses:** $$\left(x^{-1} z^{3}\right)^{-1} = x^{1} z^{-3}$$ 6. **Rewrite the entire expression:** $$x^{-3} z^{-5} \times x^{1} z^{-3} \times \frac{x^{1} y^{2} z^{4}}{x^{-3} y^{2} z^{0}}$$ 7. **Simplify the third fraction:** $$\frac{x^{1} y^{2} z^{4}}{x^{-3} y^{2} z^{0}} = x^{1 - (-3)} y^{2 - 2} z^{4 - 0} = x^{4} y^{0} z^{4} = x^{4} z^{4}$$ 8. **Multiply all parts:** $$x^{-3} z^{-5} \times x^{1} z^{-3} \times x^{4} z^{4} = x^{-3 + 1 + 4} z^{-5 - 3 + 4} = x^{2} z^{-4} = \frac{x^{2}}{z^{4}}$$ **Final answer:** $$\boxed{\frac{x^{2}}{z^{4}}}$$