1. Stating the problem: Evaluate the expression $5(2 \cdot 3 \cdot 5^4)$.\n\n2. Formula and rules: We use the order of operations (PEMDAS/BODMAS) which tells us to calculate exponents first, then multiplication.\n\n3. Calculate the exponent: $5^4 = 5 \times 5 \times 5 \times 5 = 625$.\n\n4. Substitute back: The expression becomes $5(2 \cdot 3 \cdot 625)$.\n\n5. Multiply inside the parentheses: $2 \cdot 3 = 6$, so now $5(6 \cdot 625)$.\n\n6. Multiply $6 \cdot 625 = 3750$. So the expression is $5 \times 3750$.\n\n7. Final multiplication: $5 \times 3750 = 18750$.\n\nAnswer: The value of $5(2 \cdot 3 \cdot 5^4)$ is $18750$.