To simplify the expression [tex]\( 3x \sqrt[3]{648 x^4 y^8} \)[/tex], let's go through the steps methodically.
### Step 1: Simplify the Cube Root Part Inside the Parentheses
First, we need to factor the constants and the variables inside the cube root.
#### Factorizing the Constants:
648 can be broken down into its prime factors:
[tex]\[ 648 = 2^3 \times 3^4 \][/tex]
#### Factorizing the Variables:
[tex]\[ x^4 \text{ can be written as } x^3 \times x \][/tex]
[tex]\[ y^8 \text{ can be written as } y^6 \times y^2 \][/tex]
Putting it all together, we get:
[tex]\[ 648 x^4 y^8 = 2^3 \times 3^4 \times x^3 \times x \times y^6 \times y^2 \][/tex]
### Step 2: Apply the Cube Root
Next, we take the cube root of each component:
[tex]\[ \sqrt[3]{2^3 \times 3^4 \times x^3 \times x \times y^6 \times y^2} \][/tex]
[tex]\[ = 2 \times 3^{4/3} \times x \times \sqrt[3]{3} \times y^2 \times \sqrt[3]{x \times y^2} \][/tex]
### Step 3: Combine Results with the Outer Term
Now, we multiply the simplified cube root by the outer term [tex]\( 3x \)[/tex]:
[tex]\[ 3x \times (2 \times 3^{4/3} \times x \times y^2 \times \sqrt[3]{3xy^2}) \][/tex]
[tex]\[ = 3 \times 2 \times 3^{4/3} \times x^2 \times y^2 \times \sqrt[3]{3xy^2} \][/tex]
Simplify the constants:
[tex]\[ 3 \times 2 \times 3^{4/3} = 6 \times 3^{4/3} = 18 \times 3^{1/3} \][/tex]
Note that [tex]\( 3^{1/3} \)[/tex] by itself can be considered part of the cube root factor.
Putting it all together:
[tex]\[ = 18 x^2 y^2 \sqrt[3]{3xy^2} \][/tex]
### Step 4: Identify the Correct Answer
The simplified expression is:
[tex]\[ 18 x^2 y^2 \sqrt[3]{3xy^2} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{D. \ 18 x^2 y^2 \sqrt[3]{3 x y^2}} \][/tex]
