To solve the problem using the power of a product property, let's recall how this property works. The property states:
[tex]\[
(ab)^n = a^n \cdot b^n
\][/tex]
In other words, when you have a product inside a power, you can apply the exponent to each factor separately.
Given the expression [tex]\((7y)^{1/3}\)[/tex], we apply the power of a product property as follows:
[tex]\[
(7y)^{1/3} = 7^{1/3} \cdot y^{1/3}
\][/tex]
Therefore, [tex]\((7y)^{1/3}\)[/tex] simplifies to [tex]\(7^{1/3} \cdot y^{1/3}\)[/tex].
Now, let's match this with the given answer options:
1. [tex]\(7 y^{\frac{1}{3}}\)[/tex] - This expression does not correctly apply the exponent [tex]\( \frac{1}{3} \)[/tex] to both 7 and [tex]\( y \)[/tex].
2. [tex]\(7 y^{\frac{2}{3}}\)[/tex] - This expression applies the wrong exponent to [tex]\( y \)[/tex] and does not apply the exponent to 7.
3. [tex]\(\frac{1}{7^3 y^3}\)[/tex] - This expression suggests taking the reciprocal after cubing 7 and [tex]\( y \)[/tex], which is incorrect.
4. [tex]\(7^{\frac{1}{3}} y^{\frac{1}{3}}\)[/tex] - This is the correct simplification according to the power of a product property.
Thus, the correct answer is:
[tex]\[
\boxed{7^{\frac{1}{3}} y^{\frac{1}{3}}}
\][/tex]
