To simplify the expression [tex]\(\left(3xy^4\right)^2 \left(y^2\right)^3\)[/tex], we will break it down step-by-step.
1. Simplify [tex]\(\left(3xy^4\right)^2\)[/tex]:
- Apply the power of 2 to each term inside the parentheses.
- [tex]\((3xy^4)^2 = 3^2 \cdot (x)^2 \cdot (y^4)^2\)[/tex]
- Simplify each part:
[tex]\[3^2 = 9\][/tex]
[tex]\[x^2 = x^2\][/tex]
[tex]\[(y^4)^2 = y^{4 \cdot 2} = y^8\][/tex]
- Combine these:
[tex]\[(3xy^4)^2 = 9x^2y^8\][/tex]
2. Simplify [tex]\(\left(y^2\right)^3\)[/tex]:
- Apply the power of 3 to the term inside the parentheses.
- [tex]\((y^2)^3 = y^{2 \cdot 3} = y^6\)[/tex]
3. Combine the simplified parts:
- Multiply the results from the two simplified parts:
[tex]\[9x^2y^8 \cdot y^6\][/tex]
- Use the laws of exponents to combine [tex]\(y^8\)[/tex] and [tex]\(y^6\)[/tex]:
[tex]\[y^8 \cdot y^6 = y^{8 + 6} = y^{14}\][/tex]
Therefore, the combined expression is:
[tex]\[9x^2y^{14}\][/tex]
4. Choose the correct answer:
- The final simplified expression is [tex]\(9x^2y^{14}\)[/tex], which corresponds to the second option.
Hence, the correct answer is:
[tex]\[\boxed{9x^2y^{14}}\][/tex]
