To simplify the expression [tex]\(\frac{\left(3 y^4 n^6\right)^2}{\left(y^2 n^{-3}\right)^4}\)[/tex], we should start by simplifying both the numerator and the denominator separately:
1. Simplify the numerator:
[tex]\[
\left(3 y^4 n^6\right)^2
\][/tex]
Raise each term inside the parentheses to the power of 2:
[tex]\[
(3 y^4 n^6)^2 = 3^2 \cdot (y^4)^2 \cdot (n^6)^2
\][/tex]
Calculate each of these:
[tex]\[
3^2 = 9, \quad (y^4)^2 = y^{4 \cdot 2} = y^8, \quad (n^6)^2 = n^{6 \cdot 2} = n^{12}
\][/tex]
Therefore, the numerator is:
[tex]\[
9 y^8 n^{12}
\][/tex]
2. Simplify the denominator:
[tex]\[
\left(y^2 n^{-3}\right)^4
\][/tex]
Raise each term inside the parentheses to the power of 4:
[tex]\[
(y^2 n^{-3})^4 = (y^2)^4 \cdot (n^{-3})^4
\][/tex]
Calculate each of these:
[tex]\[
(y^2)^4 = y^{2 \cdot 4} = y^8, \quad (n^{-3})^4 = n^{-3 \cdot 4} = n^{-12}
\][/tex]
Therefore, the denominator is:
[tex]\[
y^8 n^{-12}
\][/tex]
3. Combine the simplified numerator and denominator:
[tex]\[
\frac{9 y^8 n^{12}}{y^8 n^{-12}}
\][/tex]
Simplify the fraction by canceling out the common factors:
[tex]\[
\frac{9 y^8 n^{12}}{y^8 n^{-12}} = 9 \cdot \frac{y^8}{y^8} \cdot \frac{n^{12}}{n^{-12}}
\][/tex]
Since [tex]\( \frac{y^8}{y^8} = 1 \)[/tex]:
[tex]\[
9 \cdot 1 \cdot \frac{n^{12}}{n^{-12}} = 9 \cdot n^{12 - (-12)} = 9 \cdot n^{12 + 12} = 9 \cdot n^{24}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
9 n^{24}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{9 n^{24}}
\][/tex]
