To find the square root of $59684$ using the long division method, follow these steps:

---

Step 1: Pair the digits from right to left

Divide the number $59684$ into pairs of two digits starting from the right. Add extra 0s if needed.

$59 \, | \, 68 \, | \, 4$

---

Step 2: Find the largest number whose square is less than or equal to the leftmost pair

The leftmost pair is $59$.

• The largest number whose square is less than or equal to $59$ is $7$, because: $7^2 = 49 \quad \text{and} \quad 8^2 = 64 \, (> 59)$

Write $7$ as the first digit of the quotient. Subtract $49$ from $59$:

$59 - 49 = 10$

Bring down the next pair $68$, so the new dividend becomes $1068$.

---

Step 3: Double the quotient and determine the next digit

Double the current quotient $7$: $2 \times 7 = 14$

Now, find a digit $x$ such that: $14x \cdot x \leq 1068$

• If $x = 7$, $147 \cdot 7 = 1029$, which is less than $1068$.
• If $x = 8$, $148 \cdot 8 = 1184$, which exceeds $1068$.

Thus, $x = 7$. Add $7$ to the quotient and subtract $1029$ from $1068$:

$1068 - 1029 = 39$

Bring down the next pair $04$, so the new dividend becomes $3904$.

---

Step 4: Repeat the process

Double the current quotient $77$: $2 \times 77 = 154$

Now, find a digit $x$ such that: $154x \cdot x \leq 3904$

• If $x = 2$, $1542 \cdot 2 = 3084$, which is less than $3904$.
• If $x = 3$, $1543 \cdot 3 = 4629$, which exceeds $3904$.

Thus, $x = 2$. Add $2$ to the quotient and subtract $3084$ from $3904$:

$3904 - 3084 = 820$

---

Step 5: Continue for more accuracy (optional)

Bring down two zeros ($00$) to get $82000$. Repeat the process if further decimal places are needed.

---

Result

The square root of $59684$ using the long division method is approximately:

$\boxed{244}$

Would you like a detailed explanation of any step, or do you have any specific questions?

---

Related Questions

1. What is the long division method for cube roots?
2. How can you verify a square root calculation?
3. What are alternative methods to find square roots of large numbers?
4. How is the long division method related to the decimal expansion of roots?
5. Can square roots be estimated using mental math?

---

Tip

Always double-check each step in the long division process to avoid errors, especially when selecting the digits for the quotient.