1. **State the problem:** Simplify by adding like terms the expression: $$\frac{xy}{y^{-2}} - \frac{3x^{4}y^{4}}{x^{3}y} + \frac{7xy^{-2}}{xy^{-3}}$$ 2. **Recall the rules:** - When dividing powers with the same base, subtract the exponents: $$a^{m} \div a^{n} = a^{m-n}$$ - When multiplying powers with the same base, add the exponents: $$a^{m} \times a^{n} = a^{m+n}$$ - Any term with exponent zero equals 1: $$a^{0} = 1$$ 3. **Simplify each term separately:** - First term: $$\frac{xy}{y^{-2}} = xy \times y^{2} = x y^{1+2} = x y^{3}$$ - Second term: $$\frac{3x^{4}y^{4}}{x^{3}y} = 3 \times \frac{x^{4}}{x^{3}} \times \frac{y^{4}}{y^{1}} = 3 x^{4-3} y^{4-1} = 3 x^{1} y^{3} = 3 x y^{3}$$ - Third term: $$\frac{7xy^{-2}}{xy^{-3}} = 7 \times \frac{x}{x} \times \frac{y^{-2}}{y^{-3}} = 7 \times 1 \times y^{-2 - (-3)} = 7 y^{1} = 7 y$$ 4. **Rewrite the expression with simplified terms:** $$x y^{3} - 3 x y^{3} + 7 y$$ 5. **Combine like terms:** $$x y^{3} - 3 x y^{3} = (1 - 3) x y^{3} = -2 x y^{3}$$ 6. **Final simplified expression:** $$-2 x y^{3} + 7 y$$