To simplify the expression [tex]\(\ 3x \sqrt[3]{648x^4y^8} \)[/tex], follow these steps:
1. Understand the given expression:
[tex]\[
3x \sqrt[3]{648x^4y^8}
\][/tex]
Here, we have a constant [tex]\(3x\)[/tex] multiplied by the cube root of a product involving constants and variables.
2. Factor inside the cube root:
Break down the constant [tex]\(648\)[/tex] and the powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
648 = 2^3 \cdot 3^4
\][/tex]
So, the expression inside the cube root becomes:
[tex]\[
648x^4y^8 = (2^3 \cdot 3^4) \cdot x^4 \cdot y^8
\][/tex]
3. Apply the cube root to the factors:
Applying the cube root to each factor:
[tex]\[
\sqrt[3]{648x^4y^8} = \sqrt[3]{2^3 \cdot 3^4 \cdot x^4 \cdot y^8}
\][/tex]
This becomes:
[tex]\[
\sqrt[3]{2^3} \cdot \sqrt[3]{3^4} \cdot \sqrt[3]{x^4} \cdot \sqrt[3]{y^8} = 2 \cdot 3 \cdot 3^{1/3} \cdot x^{4/3} \cdot y^{8/3}
\][/tex]
4. Simplify the expression inside the cube root:
We combine the simplified terms:
[tex]\[
\sqrt[3]{648x^4y^8} = 6 \cdot 3^{1/3} \cdot x^{4/3} \cdot y^{8/3}
\][/tex]
5. Combine with the original term:
Now, multiply this result by [tex]\(3x\)[/tex]:
[tex]\[
3x \cdot 6 \cdot 3^{1/3} \cdot x^{4/3} \cdot y^{8/3} = 18x \cdot 3^{1/3} \cdot x^{4/3} \cdot y^{8/3}
\][/tex]
6. Combine like terms:
Combine the powers of [tex]\(x\)[/tex]:
[tex]\[
18x \cdot x^{4/3} = 18x^{1+4/3} = 18x^{3/3 + 4/3} = 18x^{7/3}
\][/tex]
So, the expression becomes:
[tex]\[
18 x^{7/3} \cdot y^{8/3} \cdot 3^{1/3}
\][/tex]
7. Expressing in one of the given forms:
We notice that:
[tex]\[
x^{7/3} \cdot y^{8/3} \cdot 3^{1/3} = x^{2 + 1/3} \cdot y^{2 + 2/3} \cdot 3^{1/3}
\][/tex]
This can strategically be re-distributed:
[tex]\[
18 x x^{4/3} y y^{4/3} \sqrt[3]{3}
= 18x \cdot y^2 \sqrt[3]{x^2 \cdot y^2 \cdot 3}
\][/tex]
[tex]\[
= 18x \cdot y^2 \sqrt[3]{3x^2 y^2}
\][/tex]
So, the simplified expression is:
[tex]\[
18x y^2 \sqrt[3]{3x^2 y^2}
\][/tex]
Therefore, the answer is:
[tex]\[
\boxed{D}
\][/tex]
