1. The problem is to simplify the expression $ (7y)^{\frac{1}{3}} $ using the power of a product property. 2. The power of a product property states that $ (ab)^n = a^n b^n $ for any numbers $a$, $b$, and exponent $n$. 3. Applying this property to $ (7y)^{\frac{1}{3}} $, we get: $$ (7y)^{\frac{1}{3}} = 7^{\frac{1}{3}} y^{\frac{1}{3}} $$ 4. This means the cube root of the product $7y$ is the product of the cube roots of $7$ and $y$ separately. 5. Comparing this with the options given: - $7y^{\frac{1}{3}}$ is incorrect because the 7 is not raised to the power $\frac{1}{3}$. - $7y^{\frac{2}{3}}$ is incorrect because the exponent on $y$ is wrong. - $\frac{1}{7^3 y^3}$ is incorrect because it is the reciprocal of the cube, not the cube root. - $7^{\frac{1}{3}} y^{\frac{1}{3}}$ is correct. Final answer: $$ (7y)^{\frac{1}{3}} = 7^{\frac{1}{3}} y^{\frac{1}{3}} $$