1. **State the problem:** Calculate the value of the expression $\frac{75 \cdot 175^3}{12}$.\n\n2. **Recall the order of operations:** We first calculate the exponent, then multiply, and finally divide.\n\n3. **Calculate the exponent:**\n$$175^3 = 175 \times 175 \times 175$$\nCalculate stepwise:\n$$175 \times 175 = 30625$$\n$$30625 \times 175 = 5359375$$\nSo, $$175^3 = 5359375$$.\n\n4. **Multiply by 75:**\n$$75 \times 5359375 = 401953125$$\n\n5. **Divide by 12:**\n$$\frac{401953125}{12}$$\nWe can simplify by canceling common factors if possible, but here 12 does not divide 401953125 evenly. So, perform the division:\n$$\frac{401953125}{12} = 33496093.75$$\n\n**Final answer:**\n$$33496093.75$$