1. **State the problem:** Find the square root of 4639 using the division method. 2. **Explain the division method:** This method involves grouping the digits of the number in pairs from right to left and finding the largest number whose square is less than or equal to the first group. 3. **Step 1:** Group the digits of 4639 as (46)(39). 4. **Step 2:** Find the largest number whose square is less than or equal to 46. That number is 6 because $6^2=36$ and $7^2=49$ which is greater than 46. 5. **Step 3:** Subtract $36$ from $46$ to get $10$. Bring down the next pair (39) to make $1039$. 6. **Step 4:** Double the current root (6) to get 12. Find a digit $x$ such that $12x \times x \leq 1039$. 7. **Step 5:** Try $x=8$: $128 \times 8 = 1024 \leq 1039$, try $x=9$: $129 \times 9 = 1161 > 1039$. So, $x=8$. 8. **Step 6:** Subtract $1024$ from $1039$ to get $15$. Since no more pairs to bring down, the quotient is 68 and remainder 15. 9. **Step 7:** The square root of 4639 is approximately $68$ with a remainder, so $\sqrt{4639} \approx 68$. **Final answer:** $\boxed{68}$