Question: What is the value of $\ln \sqrt[5]{e^2}$?
1. **State the problem:** We need to find the value of $\ln \sqrt[5]{e^2}$.
2. **Recall the properties of logarithms and exponents:**
- The $n$th root of a number $a$ can be written as a fractional exponent: $\sqrt[n]{a} = a^{\frac{1}{n}}$.
- The natural logarithm of an exponential function $\ln(e^x) = x$.
- The logarithm of a power: $\ln(a^b) = b \ln(a)$.
3. **Rewrite the expression inside the logarithm:**
$$\sqrt[5]{e^2} = e^{\frac{2}{5}}$$
4. **Apply the logarithm:**
$$\ln \left(e^{\frac{2}{5}}\right) = \frac{2}{5} \ln(e)$$
5. **Simplify using $\ln(e) = 1$:**
$$\frac{2}{5} \times 1 = \frac{2}{5}$$
**Final answer:**
$$\boxed{\frac{2}{5}}$$