$\sqrt[3]{v} \cdot \sqrt{v}$ =

Full solution

\sqrt[3]{v} \cdot \sqrt{v}

Given Expression: We are given the expression: $\sqrt[3]{v} \times \sqrt{v}$ $\newline$ First, we need to express both roots in terms of exponents. $\newline$ The cube root of $v$ can be written as $v^{1/3}$ and the square root of $v$ can be written as $v^{1/2}$ .
Express Roots as Exponents: Now we multiply the two expressions using the property of exponents that states when we multiply like bases, we add the exponents. $\newline$ $v^{\frac{1}{3}} \times v^{\frac{1}{2}} = v^{\frac{1}{3} + \frac{1}{2}}$
Multiply Expressions with Exponents: To add the exponents, we need a common denominator. The common denominator of $3$ and $2$ is $6$ . So we convert $\frac{1}{3}$ and $\frac{1}{2}$ to have the denominator of $6$ . $\frac{1}{3} = \frac{2}{6}$ and $\frac{1}{2} = \frac{3}{6}$
Add Exponents with Common Denominator: Now we add the exponents with the common denominator. $\newline$ $v^{\frac{2}{6} + \frac{3}{6}} = v^{\frac{5}{6}}$
Simplest Form of Expression: The expression $v^{5/6}$ is the simplest form of the product of the cube root of $v$ and the square root of $v$ .