Let's simplify the given expression step-by-step. The expression we need to simplify is:
[tex]\[
\left(6 x^2 y \right)^2 \left(y^2\right)^3
\][/tex]
### Step 1: Expand [tex]\(\left(6 x^2 y \right)^2\)[/tex]
First, look at [tex]\(\left(6 x^2 y \right)^2\)[/tex]. When squaring a product, we square each factor separately:
[tex]\[
\left(6 x^2 y \right)^2 = (6)^2 \cdot \left(x^2\right)^2 \cdot (y)^2
\][/tex]
Simplifying each part gives us:
[tex]\[
(6)^2 = 36
\][/tex]
[tex]\[
\left(x^2\right)^2 = x^{2 \times 2} = x^4
\][/tex]
[tex]\[
(y)^2 = y^2
\][/tex]
So,
[tex]\[
\left(6 x^2 y \right)^2 = 36 x^4 y^2
\][/tex]
### Step 2: Expand [tex]\(\left(y^2\right)^3\)[/tex]
Next, focus on [tex]\(\left(y^2\right)^3\)[/tex]. When raising a power to another power, we multiply the exponents:
[tex]\[
\left(y^2\right)^3 = y^{2 \times 3} = y^6
\][/tex]
### Step 3: Combine the Results
Now, multiply the results from Step 1 and Step 2:
[tex]\[
36 x^4 y^2 \cdot y^6
\][/tex]
Combine the [tex]\(y\)[/tex] terms by adding their exponents:
[tex]\[
36 x^4 y^{2+6} = 36 x^4 y^8
\][/tex]
### Conclusion
The simplified expression is:
[tex]\[
36 x^4 y^8
\][/tex]
Thus, the correct option is:
[tex]\[
\boxed{36 x^4 y^8}
\][/tex]
