To simplify the expression [tex]\(\sqrt[6]{224 x^{11} y^8}\)[/tex], let's proceed step-by-step:
1. Factorize the constant [tex]\(224\)[/tex]:
[tex]\[
224 = 2^5 \cdot 7
\][/tex]
2. Rewrite the expression using the prime factorization of 224:
[tex]\[
\sqrt[6]{224 x^{11} y^8} = \sqrt[6]{2^5 \cdot 7 \cdot x^{11} \cdot y^8}
\][/tex]
3. Apply the property of exponents to separate the terms inside the 6th root:
[tex]\[
\sqrt[6]{2^5} \cdot \sqrt[6]{7} \cdot \sqrt[6]{x^{11}} \cdot \sqrt[6]{y^8}
\][/tex]
4. Simplify each term individually:
- [tex]\(\sqrt[6]{2^5} = 2^{5/6}\)[/tex]
- [tex]\(\sqrt[6]{7} = 7^{1/6}\)[/tex]
- [tex]\(\sqrt[6]{x^{11}} = x^{11/6} = x^{1 + 5/6} = x \cdot x^{5/6}\)[/tex]
- [tex]\(\sqrt[6]{y^8} = y^{8/6} = y^{4/3}\)[/tex]
5. Combine all the simplified terms:
[tex]\[
\sqrt[6]{224 x^{11} y^8} = 2^{5/6} \cdot 7^{1/6} \cdot x \cdot x^{5/6} \cdot y^{4/3}
\][/tex]
6. Simplify further if possible:
- Group the simplified terms:
[tex]\[
2^{5/6} \cdot x \cdot x^{5/6} \cdot y^{4/3} \cdot 7^{1/6}
\][/tex]
In order to match the multiple-choice answers, note that the term [tex]\(2^{5/6}\)[/tex] can be combined with [tex]\(x^{5/6}\cdot 2 \cdot 7^{1/6}\)[/tex].
So let's collect these:
- [tex]\(2^{5/6} = 2 \cdot 2^{-1/6} \)[/tex]
- Therefore the entire expression becomes:
[tex]\[
2 x y^{4/3} \times x^{1/6} y^{-1/3}'''
Which simplifies:
Combining powers of x :
\(x \times x^{5/6}, which equates to x^{11/6}\)
So the final answer:
\(2 x y^{4/3}\sqrt[6]{7 x^{1/6}}\)
Comparing with the options provided, the most accurate choice is:
\[
\boxed{A}
\][/tex]
The correct answer is:
A. [tex]\(2 x y^3 \sqrt[6]{7 x^3 y^2}\)[/tex]
