1. **Problem statement:** Reduce the expression $5^2 \cdot (5^4)^3$ to a single power with base 5. 2. **Formula used:** When multiplying powers with the same base, add the exponents: $$a^m \cdot a^n = a^{m+n}$$ When raising a power to another power, multiply the exponents: $$(a^m)^n = a^{m \cdot n}$$ 3. **Step-by-step solution:** - First, simplify $(5^4)^3$ using the power of a power rule: $$ (5^4)^3 = 5^{4 \cdot 3} = 5^{12} $$ - Now multiply $5^2$ by $5^{12}$ using the product rule: $$ 5^2 \cdot 5^{12} = 5^{2 + 12} = 5^{14} $$ 4. **Final answer:** $$5^{14}$$ This means the original expression simplifies to $5^{14}$, a single power with base 5.