1. **State the problem:** Solve the equation $x^x = 2^{2048}$ for $x$. 2. **Recall the properties of exponents:** We know that $2^{2048}$ is a power of 2, and we want to express $x^x$ in a similar form to compare. 3. **Rewrite the right side:** $2^{2048}$ is already in exponential form. 4. **Try to express $x$ as a power of 2:** Let $x = 2^k$ for some $k$. 5. **Substitute into the equation:** $$x^x = (2^k)^{2^k} = 2^{k \cdot 2^k}$$ 6. **Set the exponents equal:** $$k \cdot 2^k = 2048$$ 7. **Solve for $k$:** Note that $2048 = 2^{11}$. Try $k = 8$: $$8 \cdot 2^8 = 8 \cdot 256 = 2048$$ This satisfies the equation. 8. **Find $x$:** $$x = 2^k = 2^8 = 256$$ **Final answer:** $$\boxed{256}$$