Answer :
To determine which expression is equivalent to [tex]\(\sqrt[3]{x^5 y}\)[/tex], let's break it down step-by-step.
First, recall that the cube root of a product can be separated into the product of the cube roots:
[tex]\[ \sqrt[3]{x^5 y} = \sqrt[3]{x^5} \cdot \sqrt[3]{y} \][/tex]
Next, we need to express each component inside the cube root as a power of [tex]\(\frac{1}{3}\)[/tex].
For [tex]\(\sqrt[3]{x^5}\)[/tex]:
[tex]\[ \sqrt[3]{x^5} = (x^5)^{\frac{1}{3}} = x^{5 \cdot \frac{1}{3}} = x^{\frac{5}{3}} \][/tex]
For [tex]\(\sqrt[3]{y}\)[/tex]:
[tex]\[ \sqrt[3]{y} = y^{\frac{1}{3}} \][/tex]
Now, combine these results:
[tex]\[ \sqrt[3]{x^5 y} = x^{\frac{5}{3}} \cdot y^{\frac{1}{3}} \][/tex]
Thus, the expression equivalent to [tex]\(\sqrt[3]{x^5 y}\)[/tex] is:
[tex]\[ x^{\frac{5}{3}} y^{\frac{1}{3}} \][/tex]
Therefore, the correct answer is:
[tex]\(\boxed{x^{\frac{5}{3}} y^{\frac{1}{3}}}\)[/tex]
First, recall that the cube root of a product can be separated into the product of the cube roots:
[tex]\[ \sqrt[3]{x^5 y} = \sqrt[3]{x^5} \cdot \sqrt[3]{y} \][/tex]
Next, we need to express each component inside the cube root as a power of [tex]\(\frac{1}{3}\)[/tex].
For [tex]\(\sqrt[3]{x^5}\)[/tex]:
[tex]\[ \sqrt[3]{x^5} = (x^5)^{\frac{1}{3}} = x^{5 \cdot \frac{1}{3}} = x^{\frac{5}{3}} \][/tex]
For [tex]\(\sqrt[3]{y}\)[/tex]:
[tex]\[ \sqrt[3]{y} = y^{\frac{1}{3}} \][/tex]
Now, combine these results:
[tex]\[ \sqrt[3]{x^5 y} = x^{\frac{5}{3}} \cdot y^{\frac{1}{3}} \][/tex]
Thus, the expression equivalent to [tex]\(\sqrt[3]{x^5 y}\)[/tex] is:
[tex]\[ x^{\frac{5}{3}} y^{\frac{1}{3}} \][/tex]
Therefore, the correct answer is:
[tex]\(\boxed{x^{\frac{5}{3}} y^{\frac{1}{3}}}\)[/tex]
