How do you calculate the square root by hand

Calculating the square root by hand can be done using several methods, such as **prime factorization**, **long division method**, or **approximation methods** like **Newton's method**. I'll explain the **long division method** because it is a reliable and widely taught method that works for both perfect and non-perfect squares.

### Long Division Method for Calculating Square Roots

This method breaks down the square root calculation into a series of simple steps, similar to long division. Here's how you can calculate the square root of a number, say 2025:

#### Step-by-Step Process:

1. **Set Up the Number in Pairs:**
   - Start from the decimal point and move both left and right, pairing digits in groups of two. For 2025, we have "20" and "25".

2. **Find the Largest Square:**
   - Look at the leftmost pair or single digit (if there is an odd number of digits). Find the largest integer whose square is less than or equal to this number. In our example, for "20", the largest integer is 4 because \(4^2 = 16\).

3. **Subtract and Bring Down:**
   - Subtract the square (16) from the leftmost pair (20), giving 4. Bring down the next pair of digits (25) next to it, making it 425.

4. **Double the Quotient and Determine the Next Digit:**
   - Take the current quotient (4), double it to get 8, and write it down. Now find a digit \(x\) such that when you calculate \( (80 + x) \times x \), it is less than or equal to 425.
   - Here, if \(x = 5\), then \( (80 + 5) \times 5 = 85 \times 5 = 425 \).

5. **Repeat the Process:**
   - Subtract 425 from 425 to get 0. If there are more digits to bring down, repeat steps 3 and 4 until you reach the desired precision.

   Since we have no more digits, the square root of 2025 is 45.

#### Summary:
The square root of 2025 using the long division method is 45. This process can be repeated for more digits or decimal places as needed.

Would you like more details or have any specific questions about this method?

### Related Questions:
1. How does the prime factorization method for finding square roots work?
2. What is Newton's method for approximating square roots?
3. How do you calculate square roots of non-perfect squares by hand?
4. What is the difference between rational and irrational square roots?
5. How can you check the accuracy of a manually calculated square root?

**Tip:** Practicing the long division method with different numbers can help you get comfortable with the process and understand the logic behind each step.

Learn how to calculate the square root of a number manually using the long division method. This detailed explanation covers step-by-step instructions for finding square roots, suitable for Grades 7-9.

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