1. The problem is to simplify the expression $5^4 \cdot 5^2$. 2. We use the rule of exponents that states when multiplying powers with the same base, we add the exponents: $$a^m \cdot a^n = a^{m+n}$$ 3. Applying this rule to our problem: $$5^4 \cdot 5^2 = 5^{4+2}$$ 4. Simplify the exponent: $$5^{4+2} = 5^6$$ 5. Calculate $5^6$: $$5^6 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 15625$$ 6. Therefore, the simplified value of $5^4 \cdot 5^2$ is $15625$.