To simplify the given expression [tex]\(\frac{3 y^4}{9 y^7}\)[/tex], we can follow these steps:
1. Factor the constants:
[tex]\[
\frac{3 y^4}{9 y^7} = \frac{3}{9} \cdot \frac{y^4}{y^7}
\][/tex]
2. Simplify the fraction of constants:
[tex]\[
\frac{3}{9} = \frac{1}{3}
\][/tex]
3. Simplify the fraction of the powers of [tex]\(y\)[/tex]:
[tex]\[
\frac{y^4}{y^7} = y^{4-7} = y^{-3}
\][/tex]
4. Combine the simplified parts:
[tex]\[
\frac{3 y^4}{9 y^7} = \frac{1}{3} \cdot y^{-3}
\][/tex]
5. Rewrite [tex]\(y^{-3}\)[/tex] as [tex]\(\frac{1}{y^3}\)[/tex]:
[tex]\[
\frac{1}{3} \cdot y^{-3} = \frac{1}{3 y^3}
\][/tex]
Thus, the simplified form of [tex]\(\frac{3 y^4}{9 y^7}\)[/tex] is [tex]\(\frac{1}{3 y^3}\)[/tex].
Therefore, the correct answer is:
[tex]\(\boxed{\frac{1}{3 y^3}}\)[/tex]
