1. **Restate the problem:** We found that $x = 256$ satisfies the equation $x^x = 2^{2048}$. 2. **Recall the form of $x$:** We expressed $x$ as a power of 2, $x = 2^k$ with $k=8$. 3. **Calculate the power:** Since $x = 2^8$, then $x^x = (2^8)^{256} = 2^{8 \times 256} = 2^{2048}$. 4. **Interpretation:** The base $x$ is $256$, and the power it is raised to is also $256$. **Final answer:** $$x = 256 \quad \text{and} \quad x \text{ is raised to the power } 256.$$