1. **State the problem:** Calculate the value of the expression $\frac{75 \cdot 175^2}{6}$. 2. **Recall the order of operations:** First, calculate the exponent, then multiply, and finally divide. 3. **Calculate the exponent:** $$175^2 = 175 \times 175 = 30625$$ 4. **Substitute back into the expression:** $$\frac{75 \cdot 30625}{6}$$ 5. **Multiply the numerator:** $$75 \cdot 30625 = 2296875$$ 6. **Divide by the denominator:** $$\frac{2296875}{6}$$ 7. **Simplify the division:** $$2296875 \div 6 = 382812.5$$ **Final answer:** $$382812.5$$