To solve the problem \( \sqrt[5]{4 x^2} \cdot \sqrt[5]{4 x^2} \), we will follow these steps:
1. Rewrite the expressions using exponent notation:
The fifth root of a number \(a\) can be written as \(a^{1/5}\). So,
[tex]\[
\sqrt[5]{4 x^2} = (4 x^2)^{1/5}
\][/tex]
2. Multiply the expressions:
According to the rules of exponents, when you multiply two expressions with the same base, you add the exponents. Therefore,
[tex]\[
\sqrt[5]{4 x^2} \cdot \sqrt[5]{4 x^2} = (4 x^2)^{1/5} \cdot (4 x^2)^{1/5} = (4 x^2)^{1/5 + 1/5} = (4 x^2)^{2/5}
\][/tex]
3. Simplify the exponent:
The exponent \(2/5\) indicates a power of 2 and a root of 5. We do not have to change this further. So our simplified expression remains:
[tex]\[
(4 x^2)^{2/5}
\][/tex]
Given the multiple choices, it becomes clear that:
- \(4 x^2\) is not simplifying correctly the power roots.
- \(\sqrt[5]{16 x^4}\) simplifies accurately but doesn't match context wise.
- \(2\left(\sqrt[5]{4 x^2}\right)\) doubles up and falls.
- \(16 x^4\) is an over syntactical amplification.
The correct choice is:
[tex]\[
\boxed{(4 x^2)^{2/5}}
\][/tex]
