1. The problem is to simplify the expression $\sqrt[25]{x^4}$.\n\n2. The general rule for radicals is $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. This means the $n$th root of $a$ to the power $m$ is equal to $a$ raised to the power $\frac{m}{n}$.\n\n3. Applying this rule, we get $$\sqrt[25]{x^4} = x^{\frac{4}{25}}.$$\n\n4. This is the simplest form since the exponent $\frac{4}{25}$ cannot be simplified further.\n\n5. Therefore, the simplified form of $\sqrt[25]{x^4}$ is $$x^{\frac{4}{25}}.$$