Question

which expression is equivalent to (sqrt3{x^5 y})?

(circ) (x^{\frac{5}{3}} y)

(circ) (x^{\frac{5}{3}} y^{\frac{1}{3}})

(circ) (x^{\frac{3}{5}} y)

(circ) (x^{\frac{3}{5}} y^3)

Explanation:

Step1: Recall radical to exponent rule

The nth root of a product \( \sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b} \) and \( \sqrt[n]{a^m}=a^{\frac{m}{n}} \). So for \( \sqrt[3]{x^5y} \), we can split it as \( \sqrt[3]{x^5}\cdot\sqrt[3]{y} \).

Step2: Apply the exponent rule

For \( \sqrt[3]{x^5} \), using \( \sqrt[n]{a^m}=a^{\frac{m}{n}} \), here \( n = 3 \), \( m = 5 \), so \( \sqrt[3]{x^5}=x^{\frac{5}{3}} \). For \( \sqrt[3]{y} \), \( n = 3 \), \( m = 1 \), so \( \sqrt[3]{y}=y^{\frac{1}{3}} \). Combining them, we get \( x^{\frac{5}{3}}y^{\frac{1}{3}} \).

Answer:

\( x^{\frac{5}{3}}y^{\frac{1}{3}} \) (the second option: \( x^{\frac{5}{3}}y^{\frac{1}{3}} \))