Let's simplify the given expression step-by-step:
Given expression:
[tex]\[
\sqrt[3]{224 \cdot x^{11} \cdot y^8}
\][/tex]
### Step 1: Prime Factorization and Separation of Terms
Start by breaking down each component inside the cube root.
- Factorizing 224:
[tex]\[ 224 = 2^5 \cdot 7 \][/tex]
So we have:
[tex]\[ 224 \cdot x^{11} \cdot y^8 = (2^5 \cdot 7) \cdot (x^{10} \cdot x) \cdot (y^6 \cdot y^2) \][/tex]
### Step 2: Separate the Terms Under the Cube Root
We can rewrite the expression:
[tex]\[
\sqrt[3]{2^5 \cdot 7 \cdot x^{10} \cdot x \cdot y^6 \cdot y^2}
\][/tex]
### Step 3: Distribute the Cube Root
Distribute the cube root across each multiplicative term:
[tex]\[
\sqrt[3]{2^5} \cdot \sqrt[3]{7} \cdot \sqrt[3]{x^{10}} \cdot \sqrt[3]{x} \cdot \sqrt[3]{y^6} \cdot \sqrt[3]{y^2}
\][/tex]
### Step 4: Simplify Each Term Within the Cube Root
- [tex]\(\sqrt[3]{2^5} = 2^{5/3}\)[/tex]
- [tex]\(\sqrt[3]{7}\)[/tex] stays as is
- [tex]\(\sqrt[3]{x^{10}} = x^{10/3}\)[/tex]
- [tex]\(\sqrt[3]{x}\)[/tex] stays as is
- [tex]\(\sqrt[3]{y^6} = y^{6/3} = y^2\)[/tex]
- [tex]\(\sqrt[3]{y^2}\)[/tex] stays as is
Combine these results:
[tex]\[
2^{5/3} x^{10/3} y^2 \cdot \sqrt[3]{7 x y^3}
\][/tex]
Now, simplify the exponents. Note that [tex]\(2^{5/3}\)[/tex] is not straightforward, but 2 can be taken out as a constant term for simplified representation.
### Step 5: Consolidate Simplified Components
We observe that:
[tex]\[
2x^2y \cdot \sqrt[3]{7xy^3}
\][/tex]
Bringing it all together, we see that the simplified form of the expression is:
[tex]\[
2 x^2 y \sqrt[3]{7 x y^3}
\][/tex]
### Conclusion:
The correct answer is:
[tex]\[
C. \quad 2 x^2 y \sqrt[3]{7 x y^3}
\][/tex]
