To solve the problem of simplifying the expression \(\frac{9 x^5 y^{16}}{45 x^5 y^4}\), we should break it down step by step:
1. Simplify the coefficients:
- The coefficients in the expression are \(9\) and \(45\).
- We simplify the fraction \(\frac{9}{45}\):
[tex]\[
\frac{9}{45} = \frac{9 \div 9}{45 \div 9} = \frac{1}{5}
\][/tex]
Hence, the fraction of the coefficients simplifies to \(\frac{1}{5}\).
2. Simplify the variables with the same base:
- For the \(x\) terms: \(\frac{x^5}{x^5}\)
[tex]\[
\frac{x^5}{x^5} = x^{5-5} = x^0 = 1
\][/tex]
Thus, the \(x\) terms cancel each other out.
- For the \(y\) terms: \(\frac{y^{16}}{y^4}\)
[tex]\[
\frac{y^{16}}{y^4} = y^{16-4} = y^{12}
\][/tex]
3. Combine the simplified results:
After simplifying both the coefficients and the variables, we get:
[tex]\[
\frac{1}{5} \cdot \frac{y^{12}}{1} = \frac{y^{12}}{5}
\][/tex]
Therefore, the simplified expression is \(\frac{y^{12}}{5}\), and the correct answer is:
\(\frac{y^{12}}{5}\)
So, the answer is:
[tex]\[\boxed{\frac{y^{12}}{5}}\][/tex]
