To simplify the given expression [tex]\(\frac{9 x^5 y^{16}}{45 x^5 y^4}\)[/tex], we need to break it down step-by-step:
1. Simplify the coefficients:
The coefficients in the expression are 9 in the numerator and 45 in the denominator. Simplify the fraction [tex]\(\frac{9}{45}\)[/tex]:
[tex]\[
\frac{9}{45} = \frac{9 \div 9}{45 \div 9} = \frac{1}{5}
\][/tex]
2. Simplify the [tex]\(x\)[/tex] terms:
The [tex]\(x\)[/tex] terms are [tex]\(x^5\)[/tex] in both the numerator and the denominator. Therefore, we can cancel these terms, as [tex]\(x^5 / x^5 = x^{5-5} = x^0 = 1\)[/tex]:
[tex]\[
\frac{x^5}{x^5} = 1
\][/tex]
3. Simplify the [tex]\(y\)[/tex] terms:
The [tex]\(y\)[/tex] terms are [tex]\(y^{16}\)[/tex] in the numerator and [tex]\(y^4\)[/tex] in the denominator. Use the properties of exponents to simplify:
[tex]\[
\frac{y^{16}}{y^4} = y^{16-4} = y^{12}
\][/tex]
Putting all the simplified parts together, we have:
[tex]\[
\frac{9 x^5 y^{16}}{45 x^5 y^4} = \frac{1}{5} \times y^{12} = \frac{y^{12}}{5}
\][/tex]
Therefore, the expression equivalent to [tex]\(\frac{9 x^5 y^{16}}{45 x^5 y^4}\)[/tex] is:
[tex]\[
\boxed{\frac{y^{12}}{5}}
\][/tex]
