To determine which expression is equivalent to [tex]\(\sqrt[3]{x^5 y}\)[/tex], let's go through the problem step-by-step.
1. Understanding the initial expression:
The given expression is the cube root of [tex]\(x^5 y\)[/tex]. In mathematical terms, this is written as:
[tex]\[
\sqrt[3]{x^5 y}
\][/tex]
2. Using the properties of exponents and radicals:
We can rewrite the cube root expression using fractional exponents. The cube root of any expression [tex]\(a\)[/tex] can be written as [tex]\(a^{\frac{1}{3}}\)[/tex].
Therefore:
[tex]\[
\sqrt[3]{x^5 y} = (x^5 y)^{\frac{1}{3}}
\][/tex]
3. Separating the components inside the parentheses:
We can use the property of exponents that states [tex]\((ab)^n = a^n b^n\)[/tex]. Applying this to our expression:
[tex]\[
(x^5 y)^{\frac{1}{3}} = (x^5)^{\frac{1}{3}} \cdot (y)^{\frac{1}{3}}
\][/tex]
4. Calculating the individual exponents:
- For [tex]\(x^5\)[/tex]:
[tex]\[
(x^5)^{\frac{1}{3}} = x^{5 \cdot \frac{1}{3}} = x^{\frac{5}{3}}
\][/tex]
- For [tex]\(y\)[/tex]:
[tex]\[
(y)^{\frac{1}{3}} = y^{\frac{1}{3}}
\][/tex]
5. Combining the results:
The expression simplifies to:
[tex]\[
x^{\frac{5}{3}} \cdot y^{\frac{1}{3}}
\][/tex]
Thus, the expression [tex]\(x^{\frac{5}{3}} y^{\frac{1}{3}}\)[/tex] is equivalent to [tex]\(\sqrt[3]{x^5 y}\)[/tex].
Therefore, the correct answer is:
[tex]\[
x^{\frac{5}{3}} y^{\frac{1}{3}}
\][/tex]
