Find the square root of 129636 and explain each step

To find the square root of 129636, we can follow these steps:

### Step 1: Estimating the Square Root
First, we estimate the square root by finding two perfect squares close to 129636.

- \( \sqrt{121000} \approx 348 \)  
- \( \sqrt{144000} \approx 379 \)

Since 129636 lies between these two numbers, we know the square root will be between 348 and 379.

### Step 2: Long Division Method
We now use the long division method to find the precise value of the square root.

#### 1. Group the digits in pairs, starting from the decimal point.
   For 129636, group it as \( 12 | 96 | 36 \).

#### 2. Find the largest square number less than or equal to 12.
   \( 3^2 = 9 \) is the closest perfect square.
   Write down 3 as the first digit of the square root.

   Subtract \( 9 \) from \( 12 \):
   \[
   12 - 9 = 3
   \]

   Bring down the next group of digits (96), so we have 396.

#### 3. Double the current quotient (3) to get 6. Place it as the first part of the new divisor.
   Now we need to find a digit \( x \) such that \( 6x \times x \leq 396 \).
   Try \( x = 6 \):
   \[
   66 \times 6 = 396
   \]
   Subtract \( 396 - 396 = 0 \).

   Now bring down the next group of digits (36), so we have 36.

#### 4. Double the quotient again (36), which gives 72.
   Now we find a digit \( x \) such that \( 72x \times x \leq 36 \).
   Try \( x = 0 \):
   \[
   720 \times 0 = 0
   \]
   Subtract \( 36 - 0 = 36 \).

At this point, we stop as we have obtained enough accuracy to confirm that:

### Final Result:
\[
\sqrt{129636} = 360
\]

### Conclusion:
The square root of 129636 is exactly 360.

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Do you want further details or have any questions?

Here are 5 related questions:
1. What is the square root of 12321 using the long division method?
2. How can you estimate square roots quickly for large numbers?
3. What are some other methods to find square roots without using a calculator?
4. How does the long division method compare to using a calculator for square roots?
5. Can you explain how to find cube roots using a similar method?

**Tip**: When estimating square roots, it's useful to memorize the squares of numbers from 1 to 20 for quick reference.

Learn how to find the square root of 129636 using the long division method in a step-by-step guide. This problem covers key concepts like square root estimation and the long division process. Suitable for students in Grades 8-10.

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