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 the radical to exponent rule

The formula for converting a radical to an exponent is $\sqrt[n]{a^m}=a^{\frac{m}{n}}$. For a product inside the radical, $\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}$ (where $a$ and $b$ are non - negative real numbers and $n$ is a positive integer greater than 1). So, for $\sqrt[3]{x^{5}y}$, we can rewrite it as $\sqrt[3]{x^{5}}\cdot\sqrt[3]{y}$.

Step2: Apply the radical - exponent rule to each term

For $\sqrt[3]{x^{5}}$, using the rule $\sqrt[n]{a^m}=a^{\frac{m}{n}}$ with $n = 3$ and $m = 5$, we get $x^{\frac{5}{3}}$. For $\sqrt[3]{y}$, using the rule with $n = 3$ and $m = 1$ (since $y=y^{1}$), we get $y^{\frac{1}{3}}$.

Step3: Combine the terms

Multiplying the two results together, we have $x^{\frac{5}{3}}y^{\frac{1}{3}}$.

Answer:

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