Multiply and Simplify: (⁵√√3) × (⁵√√3) Radical Expression

To solve the problem $\sqrt[5]{\sqrt{3}} \cdot \sqrt[5]{\sqrt{3}} =$, we follow these steps:

Step 1: Express each root using exponents.
$\sqrt[5]{\sqrt{3}}$ can be rewritten as $(3^{1/2})^{1/5}$, which simplifies to $3^{1/10}$ using the law $(a^m)^n = a^{m \cdot n}$.

Step 2: Multiply the expressions.
We have $(3^{1/10}) \cdot (3^{1/10})$. According to the laws of exponents, $a^m \cdot a^n = a^{m+n}$. Thus, the expression becomes $3^{1/10 + 1/10} = 3^{2/10} = 3^{1/5}$.

Step 3: Convert back to a root, if necessary.
The expression $3^{1/5}$ corresponds to $\sqrt[5]{3}$.

Therefore, the expression $\sqrt[5]{\sqrt{3}} \cdot \sqrt[5]{\sqrt{3}}$ simplifies to $3^{1/5}$, which is equivalent to $\sqrt[5]{9}$.

To match with the given choices, observe that $3^{1/5}$ can also be expressed as $\sqrt[10]{9}$ because $3^{1/5} = (3^2)^{1/10}$, which equals to $\sqrt[10]{9}$.

The correct answer is choice 4, $\sqrt[10]{9}$.