1. Problem: Find $x$ in the equation $x^x = n^n$. 2. Formula and rules: When bases and exponents are equal, the expressions are equal. Here, $x^x = n^n$ implies $x = n$ because the function $f(t) = t^t$ is one-to-one for $t > 0$. 3. Intermediate work: Since $x^x = n^n$, the simplest solution is $x = n$. 4. Explanation: The function $t^t$ is strictly increasing for $t > 0$, so if $x^x = n^n$, then $x = n$. Final answer: $x = n$.