1. **State the problem:** Simplify the expression $$x^{-2}y^{-2}(x^2 y - 1 - y^2 x^{-1})$$. 2. **Recall the rules:** - When multiplying powers with the same base, add exponents: $$a^m \cdot a^n = a^{m+n}$$. - Negative exponents mean reciprocal: $$a^{-n} = \frac{1}{a^n}$$. - Distribute multiplication over addition/subtraction. 3. **Distribute $$x^{-2}y^{-2}$$ over each term inside the parentheses:** $$x^{-2}y^{-2} \cdot x^2 y - x^{-2}y^{-2} \cdot 1 - x^{-2}y^{-2} \cdot y^2 x^{-1}$$ 4. **Simplify each term using exponent rules:** - First term: $$x^{-2}y^{-2} \cdot x^2 y = x^{-2+2} y^{-2+1} = x^0 y^{-1} = y^{-1}$$ - Second term: $$x^{-2}y^{-2} \cdot 1 = x^{-2} y^{-2}$$ - Third term: $$x^{-2}y^{-2} \cdot y^2 x^{-1} = x^{-2-1} y^{-2+2} = x^{-3} y^0 = x^{-3}$$ 5. **Rewrite the expression with simplified terms:** $$y^{-1} - x^{-2} y^{-2} - x^{-3}$$ 6. **Express negative exponents as fractions:** $$\frac{1}{y} - \frac{1}{x^2 y^2} - \frac{1}{x^3}$$ **Final answer:** $$\frac{1}{y} - \frac{1}{x^2 y^2} - \frac{1}{x^3}$$