1. **State the problem:** Calculate $50\sqrt{900500}$ using the long division method for square roots. 2. **Recall the long division method:** This method finds the square root by grouping digits in pairs from the decimal point and finding digits of the root step-by-step. 3. **Group the number:** Write $900500$ as pairs from right: $9\,00\,50\,0$ (adding a zero to make pairs even). 4. **Find the largest square less than or equal to the first group (9):** $3^2=9$. 5. **Subtract and bring down next pair:** $9-9=0$, bring down $00$, remainder is $0$. 6. **Double the root so far (3):** $2\times3=6$, find digit $x$ such that $6x \times x \leq 0 0$; $x=0$. 7. **Repeat for next pairs:** Bring down $50$, new divisor $60$, find $x$ with $60x \times x \leq 50$; $x=0$. 8. **Bring down next pair (0):** New divisor $600$, find $x$ with $600x \times x \leq 500$; $x=0$. 9. **Root found so far:** $3000$. 10. **Calculate $\sqrt{900500} \approx 3000$ (approximate). 11. **Multiply by 50:** $50 \times 3000 = 150000$. **Final answer:** $50\sqrt{900500} \approx 150000$