1. **State the problem:** Simplify the expression $9 x^{5} y^{2} x - 2 x^{-2} y^{-2}$. 2. **Rewrite the expression:** Combine like bases with exponents where possible. $$9 x^{5} y^{2} x = 9 x^{5+1} y^{2} = 9 x^{6} y^{2}$$ So the expression becomes: $$9 x^{6} y^{2} - 2 x^{-2} y^{-2}$$ 3. **Important rules:** - When multiplying powers with the same base, add exponents: $a^{m} \cdot a^{n} = a^{m+n}$. - Negative exponents mean reciprocal: $a^{-m} = \frac{1}{a^{m}}$. 4. **Simplify the expression:** The expression is already simplified as a subtraction of two terms: $$9 x^{6} y^{2} - 2 x^{-2} y^{-2}$$ If desired, rewrite negative exponents as fractions: $$9 x^{6} y^{2} - 2 \frac{1}{x^{2} y^{2}} = 9 x^{6} y^{2} - \frac{2}{x^{2} y^{2}}$$ 5. **Final answer:** $$9 x^{6} y^{2} - \frac{2}{x^{2} y^{2}}$$ This is the simplified form of the given expression.