1. **State the problem:** Simplify the expression $$\frac{x^{-3} y^{3} x^{4}}{x^{-2} x^{5} y^{3}}$$. 2. **Recall the exponent rules:** - When multiplying like bases, add exponents: $$a^{m} \cdot a^{n} = a^{m+n}$$. - When dividing like bases, subtract exponents: $$\frac{a^{m}}{a^{n}} = a^{m-n}$$. - Any base to the zero power is 1: $$a^{0} = 1$$. 3. **Simplify numerator:** $$x^{-3} x^{4} y^{3} = x^{-3+4} y^{3} = x^{1} y^{3} = x y^{3}$$. 4. **Simplify denominator:** $$x^{-2} x^{5} y^{3} = x^{-2+5} y^{3} = x^{3} y^{3}$$. 5. **Rewrite the fraction:** $$\frac{x y^{3}}{x^{3} y^{3}}$$. 6. **Divide like bases:** - For $x$: $$\frac{x}{x^{3}} = x^{1-3} = x^{-2}$$. - For $y$: $$\frac{y^{3}}{y^{3}} = y^{3-3} = y^{0} = 1$$. 7. **Final simplified expression:** $$x^{-2} \cdot 1 = x^{-2} = \frac{1}{x^{2}}$$. **Answer:** $$\boxed{\frac{1}{x^{2}}}$$