Sure! Let's analyze the expression [tex]\(\sqrt[3]{x^5 y}\)[/tex].
1. Understanding cube roots:
- The cube root of any expression [tex]\(\sqrt[3]{A}\)[/tex] can be written as [tex]\(A^{\frac{1}{3}}\)[/tex].
2. Apply the cube root:
- Given the expression [tex]\(\sqrt[3]{x^5 y}\)[/tex], we can rewrite this as [tex]\((x^5 y)^{\frac{1}{3}}\)[/tex].
3. Distribute the cube root exponent:
- When we have a product inside an exponent, we can apply the exponent to each individual factor inside the product:
[tex]\[
(x^5 y)^{\frac{1}{3}} = x^{5 \cdot \frac{1}{3}} \cdot y^{1 \cdot \frac{1}{3}}
\][/tex]
4. Simplify the exponents:
- Simplify each exponent separately:
[tex]\[
x^{5 \cdot \frac{1}{3}} = x^{\frac{5}{3}}
\][/tex]
[tex]\[
y^{1 \cdot \frac{1}{3}} = y^{\frac{1}{3}}
\][/tex]
5. Combine the results:
- Putting it all together, we get:
[tex]\[
x^{\frac{5}{3}} y^{\frac{1}{3}}
\][/tex]
Therefore, the expression [tex]\(x^{\frac{5}{3}} y^{\frac{1}{3}}\)[/tex] is equivalent to [tex]\(\sqrt[3]{x^5 y}\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{x^{\frac{5}{3}} y^{\frac{1}{3}}}
\][/tex]
