1. The problem is to simplify the expression $5^4 \cdot 5$. 2. Recall the rule for multiplying powers with the same base: $a^m \cdot a^n = a^{m+n}$. 3. Applying this rule, we add the exponents: $$5^4 \cdot 5^1 = 5^{4+1} = 5^5$$ 4. Therefore, the simplified expression is $5^5$. 5. Among the given options, $5^5$ is the correct answer.