To simplify the expression [tex]\(\left(6 x^2 y\right)^2 \left(y^2\right)^3\)[/tex], we will begin by working step-by-step with each part of the expression.
### Step 1: Simplify [tex]\(\left(6 x^2 y\right)^2\)[/tex]
First, let's break down this part:
[tex]\[
\left(6 x^2 y\right)^2
\][/tex]
When we square a product inside the parentheses, each factor inside is raised to the power of 2:
[tex]\[
= 6^2 \cdot \left(x^2\right)^2 \cdot y^2
\][/tex]
Calculate each component:
[tex]\[
6^2 = 36
\][/tex]
[tex]\[
\left(x^2\right)^2 = x^{2 \times 2} = x^4
\][/tex]
[tex]\[
y^2 = y^2
\][/tex]
Thus,
[tex]\[
\left(6 x^2 y\right)^2 = 36 x^4 y^2
\][/tex]
### Step 2: Simplify [tex]\(\left(y^2\right)^3\)[/tex]
Next, we simplify this part:
[tex]\[
\left(y^2\right)^3
\][/tex]
Raise [tex]\(y^2\)[/tex] to the power of 3:
[tex]\[
= y^{2 \times 3} = y^6
\][/tex]
### Step 3: Combine the simplified parts
Now, we combine the results from steps 1 and 2:
[tex]\[
(36 x^4 y^2) \cdot (y^6)
\][/tex]
When multiplying terms with the same base, add the exponents:
[tex]\[
= 36 x^4 y^{2 + 6} = 36 x^4 y^8
\][/tex]
### Conclusion
Hence, the correct simplification of the expression [tex]\(\left(6 x^2 y\right)^2 \left(y^2\right)^3\)[/tex] is:
[tex]\[
36 x^4 y^8
\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{36 x^4 y^8}
\][/tex]
