Let's simplify the given expression step-by-step:
[tex]\[
8 x^2 \sqrt[3]{375 x} + 2 \sqrt[3]{3 x^7}
\][/tex]
### Step 1: Simplify Each Term
First, we handle each part separately:
#### Term 1: [tex]\(8 x^2 \sqrt[3]{375 x}\)[/tex]
1. Simplify [tex]\(375 x\)[/tex]:
[tex]\[
375x = 375 \cdot x
\][/tex]
2. Find the cube root of [tex]\(375\)[/tex]:
[tex]\[
\sqrt[3]{375} = \sqrt[3]{125 \cdot 3} = \sqrt[3]{125} \cdot \sqrt[3]{3} = 5 \cdot \sqrt[3]{3}
\][/tex]
3. Therefore:
[tex]\[
\sqrt[3]{375 x} = \sqrt[3]{375} \cdot \sqrt[3]{x} = 5 \cdot \sqrt[3]{3} \cdot \sqrt[3]{x}
\][/tex]
4. Substitute back into the term [tex]\(8 x^2 \sqrt[3]{375 x}\)[/tex]:
[tex]\[
8 x^2 \cdot (5 \sqrt[3]{3} \sqrt[3]{x}) = 8 x^2 \cdot 5 \sqrt[3]{3x} = 40 x^2 \sqrt[3]{3x}
\][/tex]
#### Term 2: [tex]\(2 \sqrt[3]{3 x^7}\)[/tex]
1. Simplify [tex]\(3 x^7\)[/tex]:
[tex]\[
3 x^7 = 3 \cdot x^7
\][/tex]
2. Find the cube root of [tex]\(3 x^7\)[/tex]:
[tex]\[
\sqrt[3]{3 x^7} = \sqrt[3]{3} \cdot \sqrt[3]{x^7} = \sqrt[3]{3} \cdot x^{7/3}
\][/tex]
3. Therefore:
[tex]\[
\sqrt[3]{3 x^7} = \sqrt[3]{3} \cdot x^{7/3}
\][/tex]
4. Substitute back into the term [tex]\(2 \sqrt[3]{3 x^7}\)[/tex]:
[tex]\[
2 \sqrt[3]{3 x^7} = 2 \cdot \sqrt[3]{3} \cdot x^{7/3}
\][/tex]
### Step 2: Combine the Simplified Terms
Now, add the two simplified terms together:
[tex]\[
40 x^2 \sqrt[3]{3x} + 2 \sqrt[3]{3} x^{7/3}
\][/tex]
Note that both terms contain [tex]\(\sqrt[3]{3}\)[/tex] and can potentially be factored out if appropriately expressed in like terms:
[tex]\[
40 x^2 \sqrt[3]{3x} + 2 \sqrt[3]{3} x^{7/3}
\][/tex]
Express [tex]\(40 x^2 \sqrt[3]{3x}\)[/tex] as:
[tex]\[
40 \sqrt[3]{3} x^{2 + 1/3} = 40 \sqrt[3]{3} x^{7/3}
\][/tex]
So the full expression now becomes:
[tex]\[
42 x^{7/3} \sqrt[3]{3}
\][/tex]
Written in a more simplified form, this evaluation corresponds to one of the provided choices.
[tex]\[
42 x^{(6/3 + 1/3)} \sqrt[3]{3x}
\][/tex]
The correct answer among the options given is:
[tex]\(\boxed{B. \ 42 x^2 \sqrt[3]{3 x}}\)[/tex]
