To simplify the expression [tex]\(3x \sqrt[3]{648x^4y^8}\)[/tex], we should follow these steps:
### Step 1: Simplify the Expression Inside the Cube Root
First, let's simplify [tex]\(648x^4y^8\)[/tex] inside the cube root:
[tex]\[648x^4y^8\][/tex]
Factor [tex]\(648\)[/tex] into its prime factors:
[tex]\[648 = 2^3 \times 3^4\][/tex]
So, the expression inside the cube root is:
[tex]\[2^3 \times 3^4 \times x^4 \times y^8\][/tex]
### Step 2: Take the Cube Root of Each Factor
Apply the cube root to each factor:
[tex]\[
\sqrt[3]{2^3 \times 3^4 \times x^4 \times y^8}
\][/tex]
Recall:
[tex]\[
\sqrt[3]{a^3} = a
\][/tex]
Thus:
[tex]\[
\sqrt[3]{2^3} = 2
\][/tex]
For [tex]\(3^4\)[/tex], we break it into:
[tex]\[
3^4 = 3^3 \times 3
\][/tex]
So:
[tex]\[
\sqrt[3]{3^4} = \sqrt[3]{3^3 \times 3} = 3 \sqrt[3]{3}
\][/tex]
For [tex]\(x^4\)[/tex], break it into:
[tex]\[
x^4 = x^3 \times x
\][/tex]
Thus:
[tex]\[
\sqrt[3]{x^4} = \sqrt[3]{x^3 \times x} = x \sqrt[3]{x}
\][/tex]
For [tex]\(y^8\)[/tex], break it into:
[tex]\[
y^8 = (y^3)^2 \times y^2
\][/tex]
Thus:
[tex]\[
\sqrt[3]{y^8} = \sqrt[3]{(y^3)^2 \times y^2} = y^2 \sqrt[3]{y^2}
\][/tex]
Putting it all together, we get:
[tex]\[
\sqrt[3]{2^3 \times 3^4 \times x^4 \times y^8} = 2 \times 3 \sqrt[3]{3} \times x \sqrt[3]{x} \times y^2 \sqrt[3]{y^2}
\][/tex]
Combining the terms inside the cube root, we have:
[tex]\[
2 \times 3 \times x \times y^2 \sqrt[3]{3xy^2}
\][/tex]
This simplifies to:
[tex]\[
6xy^2 \sqrt[3]{3xy^2}
\][/tex]
### Step 3: Multiply by 3x
Now, multiply the simplified cube root expression by [tex]\(3x\)[/tex]:
[tex]\[
3x \times 6xy^2 \sqrt[3]{3xy^2} = 18x^2y^2\sqrt[3]{3xy^2}
\][/tex]
So the simplified expression is:
[tex]\[
18x^2y^2\sqrt[3]{3xy^2}
\][/tex]
Match this to the given choices; it corresponds to:
[tex]\[
\boxed{C. 18 x^2 y^2 \sqrt[3]{3 x y^2}}
\][/tex]
