To simplify the given expression [tex]\(\left(5 x y^5\right)^2\left(y^3\right)^4\)[/tex], we can follow these steps:
1. Simplify [tex]\(\left(5 x y^5\right)^2\)[/tex]:
Raise [tex]\(5 x y^5\)[/tex] to the power of 2. This means we need to square each of the factors inside the parentheses.
[tex]\[
\left(5 x y^5\right)^2 = (5)^2 \cdot (x)^2 \cdot (y^5)^2
\][/tex]
Calculate each part separately:
[tex]\[
(5)^2 = 25
\][/tex]
[tex]\[
(x)^2 = x^2
\][/tex]
[tex]\[
(y^5)^2 = y^{5 \cdot 2} = y^{10}
\][/tex]
Combine these results:
[tex]\[
\left(5 x y^5\right)^2 = 25 x^2 y^{10}
\][/tex]
2. Simplify [tex]\(\left(y^3\right)^4\)[/tex]:
Raise [tex]\(y^3\)[/tex] to the power of 4. Multiply the exponent inside the parentheses by the exponent outside.
[tex]\[
\left(y^3\right)^4 = y^{3 \cdot 4} = y^{12}
\][/tex]
3. Multiply the simplified parts:
Combine the simplified expressions [tex]\(\left(5 x y^5\right)^2 = 25 x^2 y^{10}\)[/tex] and [tex]\(\left(y^3\right)^4 = y^{12}\)[/tex].
[tex]\[
25 x^2 y^{10} \cdot y^{12}
\][/tex]
Since the bases of the [tex]\(y\)[/tex] terms are the same, we add the exponents:
[tex]\[
25 x^2 y^{10 + 12} = 25 x^2 y^{22}
\][/tex]
Therefore, the correct simplification of the expression [tex]\(\left(5 x y^5\right)^2\left(y^3\right)^4\)[/tex] is:
[tex]\[
\boxed{25 x^2 y^{22}}
\][/tex]
