Answer :
To determine which expression is equivalent to \(\sqrt[3]{x^5 y}\), we can follow the properties of exponents and radicals.
1. Recall the property of radicals which states that \(\sqrt[n]{a} = a^{1/n}\). Applying this to \(\sqrt[3]{x^5 y}\), we can rewrite it as \((x^5 y)^{1/3}\).
2. According to the properties of exponents, when you have a product within a radical, you can distribute the exponent to each term. Thus,
[tex]\[ (x^5 y)^{1/3} = (x^5)^{1/3} \cdot (y)^{1/3} \][/tex]
3. Now evaluate each term separately:
- For \(x^5\), we use the rule \((a^m)^n = a^{m \cdot n}\):
[tex]\[ (x^5)^{1/3} = x^{5 \cdot 1/3} = x^{5/3} \][/tex]
- For \(y\), applying the exponent of \(1/3\), we get:
[tex]\[ y^{1/3} \][/tex]
4. Combine the results from steps 3:
[tex]\[ (x^5)^{1/3} \cdot (y)^{1/3} = x^{5/3} \cdot y^{1/3} \][/tex]
Thus, the expression that is equivalent to \(\sqrt[3]{x^5 y}\) is:
[tex]\[ x^{\frac{5}{3}} y^{\frac{1}{3}} \][/tex]
Therefore, the correct choice is:
[tex]\[ x^{\frac{5}{3}} y^{\frac{1}{3}} \][/tex]
1. Recall the property of radicals which states that \(\sqrt[n]{a} = a^{1/n}\). Applying this to \(\sqrt[3]{x^5 y}\), we can rewrite it as \((x^5 y)^{1/3}\).
2. According to the properties of exponents, when you have a product within a radical, you can distribute the exponent to each term. Thus,
[tex]\[ (x^5 y)^{1/3} = (x^5)^{1/3} \cdot (y)^{1/3} \][/tex]
3. Now evaluate each term separately:
- For \(x^5\), we use the rule \((a^m)^n = a^{m \cdot n}\):
[tex]\[ (x^5)^{1/3} = x^{5 \cdot 1/3} = x^{5/3} \][/tex]
- For \(y\), applying the exponent of \(1/3\), we get:
[tex]\[ y^{1/3} \][/tex]
4. Combine the results from steps 3:
[tex]\[ (x^5)^{1/3} \cdot (y)^{1/3} = x^{5/3} \cdot y^{1/3} \][/tex]
Thus, the expression that is equivalent to \(\sqrt[3]{x^5 y}\) is:
[tex]\[ x^{\frac{5}{3}} y^{\frac{1}{3}} \][/tex]
Therefore, the correct choice is:
[tex]\[ x^{\frac{5}{3}} y^{\frac{1}{3}} \][/tex]
