Square Root with Division Method: A Step-by-Step Guide

The long division method is an effective way to manually find the square root of a number, particularly useful for both perfect and non-perfect squares. This method involves a series of steps that simplify the process, allowing for accurate results without the need for a calculator.

Steps to Find the Square Root

1. Group the digits in pairs, starting from the decimal point (or the rightmost digit for whole numbers). Each pair is called a period.
2. Find the largest number whose square is less than or equal to the first period. This number is the divisor and the quotient.
3. Subtract the square of the divisor from the first period and bring down the next period to the right of the remainder.
4. Double the quotient and enter it as the new divisor, leaving a blank digit to the right. Find a digit to fill the blank so that when the new divisor is multiplied by this digit, the product is less than or equal to the current dividend.
5. Repeat the steps of bringing down pairs of digits and finding new divisors until you reach the desired accuracy.

Example Calculation

Let's find the square root of 484 using the long division method:

1. Pair the digits: \( \overline{4} \ \overline{84} \).
2. Find the largest square less than or equal to 4, which is 2 (since \(2^2 = 4\)). Write 2 as the quotient.
3. Subtract \(4\) from \(4\), giving a remainder of \(0\). Bring down the next period (84), making the new dividend 84.
4. Double the quotient (2), giving 4. Now, find a digit \(x\) such that \(4x \times x \leq 84\). The digit is 2 because \(42 \times 2 = 84\).
5. The remainder is 0, and we have no more digits to bring down. Thus, \( \sqrt{484} = 22 \).

Additional Examples

Here are a few more examples to illustrate the process:

• Find the square root of 68: \( \sqrt{68} \approx 8.246 \).
• Find the square root of 784: \( \sqrt{784} = 28 \).
• Find the square root of 5329: \( \sqrt{5329} = 73 \).

Finding Square Roots of Non-Perfect Squares

For non-perfect squares, the method remains the same but continues to the desired decimal places. For example, the square root of 128 by the long division method is approximately 11.313.

The long division method is not only accurate but also provides a deep understanding of the concept of square roots and how they are derived, making it a valuable tool for learning and teaching mathematics.

For a visual explanation, check out this on the long division method for finding square roots.

Start practicing this method to enhance your mathematical skills and understanding!

The division method for finding square roots is a systematic and precise technique, particularly useful for both perfect and non-perfect squares. This method involves several steps that simplify the process and ensure accurate results, making it a valuable tool in mathematics. Here's a detailed introduction to this method:

The square root of a number is a value that, when multiplied by itself, gives the original number. The division method breaks down the process into manageable steps, making it easier to compute the square root manually. This method is particularly effective for large numbers and provides a clear understanding of the calculation process.

1. Pairing the Digits: Begin by grouping the digits of the number into pairs, starting from the decimal point. For whole numbers, start from the rightmost digit. Each pair is processed separately.
2. Initial Division: Find the largest number whose square is less than or equal to the first pair or single digit on the left. This number is the initial quotient and divisor.
3. Subtraction and Bring Down: Subtract the square of the initial divisor from the first pair and bring down the next pair of digits. This forms the new dividend.
4. Double the Quotient: Double the current quotient and write it down with a blank space next to it. This new number will be part of the next divisor.
5. Find the Next Digit: Identify the largest possible digit to fill the blank space in the new divisor, such that when the new divisor is multiplied by this digit, the product is less than or equal to the current dividend.
6. Repeat the Process: Repeat the steps of subtraction, bringing down the next pair of digits, doubling the quotient, and finding the next digit until all pairs of digits have been processed.

Let's illustrate this with an example:

1. Consider the number 484. Pair the digits: \( \overline{4} \ \overline{84} \).
2. Find the largest square less than or equal to 4, which is 2 (since \(2^2 = 4\)). Write 2 as the quotient and the divisor.
3. Subtract \(4\) from \(4\), giving a remainder of \(0\). Bring down the next period (84), making the new dividend 84.
4. Double the quotient (2), giving 4. Now, find a digit \(x\) such that \(4x \times x \leq 84\). The digit is 2 because \(42 \times 2 = 84\).
5. The remainder is 0, and we have no more digits to bring down. Thus, \( \sqrt{484} = 22 \).

This method not only helps in finding square roots without a calculator but also enhances understanding of the underlying mathematical concepts. It's a valuable skill for students and professionals alike.

The division method for finding the square root of a number involves several systematic steps. Here's a detailed guide to help you through the process:

1. Pair the Digits:

   Start by pairing the digits of the number from right to left. For example, for the number 2025, the pairs will be (20)(25).

2. Find the Largest Square:

   Identify the largest square less than or equal to the first pair (or single digit). This becomes the first digit of your answer. Subtract this square from the first pair and bring down the next pair. For example, if the first pair is 20, the largest square is 16 (4²), so the first digit is 4.

3. Double the Result:

   Double the first digit of the result and place it as the divisor for the next step. Continue the process with the new pairs.

4. Guess and Multiply:

   Find a digit that, when appended to the divisor and multiplied by itself, gives a product less than or equal to the current dividend. Subtract this product from the current dividend and bring down the next pair of digits.

5. Repeat:

   Repeat the process until all pairs have been processed. The quotient will be the square root of the number.

Let's apply these steps with an example:

1. Find the square root of 2025.
2. Pair the digits: (20)(25).
3. The largest square less than 20 is 16 (4²), so write 4. Subtract 16 from 20 to get 4. Bring down the next pair to make it 425.
4. Double the 4 to get 8. Now, guess a digit X such that 8X * X is less than or equal to 425. The suitable X here is 5 because 85 * 5 = 425.
5. Subtract 425 from 425 to get 0. Thus, the square root of 2025 is 45.

Below are some examples demonstrating the process of finding square roots using the long division method:

Example 1: Square Root of 169

1. Group the digits into pairs starting from the decimal point. Here, 169 becomes (1)(69).
2. Find the largest number whose square is less than or equal to the first group. In this case, the largest number is 1 (since \(1^2 = 1\)). Place 1 as the first digit of the quotient.
3. Subtract \(1^2\) from the first group, bringing down the next pair. So, \(1-1=0\) and the next pair is 69, making the new dividend 069.
4. Double the quotient (which is currently 1) and write it as the new divisor's first digit: 2_.
5. Find a digit to complete the divisor so that when multiplied by itself, the product is less than or equal to the new dividend. Here, 26 * 6 = 156. The closest possible is 24 * 4 = 96, which is greater. So we choose 23 * 3 = 69, which fits perfectly.
6. Subtract 69 from 69 to get 0. The quotient is 13, and since there's no remainder, \( \sqrt{169} = 13 \).

Example 2: Square Root of 72

1. Group the digits into pairs starting from the decimal point. Here, 72 becomes (72).
2. Find the largest number whose square is less than or equal to the first group. The largest number is 8 (since \(8^2 = 64\)). Place 8 as the first digit of the quotient.
3. Subtract \(64\) from \(72\), leaving a remainder of \(8\).
4. Double the quotient (which is currently 8) and write it as the new divisor's first digit: 16_.
5. Since we are only considering up to the first decimal place, find the appropriate digit to complete the divisor: 160 * 0 = 0. The closest match is 160 * 5 = 800, which is greater. So we choose 160 * 4 = 640, which fits closely but not perfectly.
6. After calculating further, the final approximation of the square root of 72 is \( \approx 8.48 \).

Example 3: Square Root of 225

1. Group the digits into pairs starting from the decimal point. Here, 225 becomes (2)(25).
2. Find the largest number whose square is less than or equal to the first group. The largest number is 1 (since \(1^2 = 1\)). Place 1 as the first digit of the quotient.
3. Subtract \(1^2\) from the first group, bringing down the next pair. So, \(2-1=1\) and the next pair is 25, making the new dividend 125.
4. Double the quotient (which is currently 1) and write it as the new divisor's first digit: 2_.
5. Find a digit to complete the divisor so that when multiplied by itself, the product is less than or equal to the new dividend. Here, 22 * 2 = 44. The closest possible is 22 * 2 = 44, which fits perfectly.
6. Subtract 44 from 125 to get 81. The next digit is 5.
7. Double the quotient to get 15 and find a digit to complete the divisor. 155 * 5 = 775, closest possible is 155 * 5 = 775, fits perfectly. So the square root is \( \sqrt{225} = 15 \).

The process of finding the square root of decimal numbers using the long division method involves several steps. Here is a detailed, step-by-step guide to help you understand this method.

1. Write the number in decimal form and group the digits in pairs, starting from the decimal point and moving both to the left and the right. For example, consider the number 24.01. Group the digits as \( \overline{24}.\overline{01} \).

2. Find the largest number whose square is less than or equal to the first group (the integer part). In this case, the largest number is 4, because \( 4^2 = 16 \) and is less than 24. Place 4 as the first digit of the quotient.

3. Subtract the square of the quotient (4) from the first group and bring down the next group of digits (01) next to the remainder. The new dividend is 801.

4. Double the quotient (4) and write it below the new dividend, leaving a blank space next to it (i.e., write 8_).

5. Determine the largest digit (X) to fill the blank such that \( (80 + X) \times X \leq 801 \). Here, 9 is the largest digit, as \( 89 \times 9 = 801 \).

6. Write 9 next to 4 in the quotient, giving 4.9. Since there are no more digits left, the process stops here, and the square root of 24.01 is 4.9.

Another example can help clarify the process:

1. Consider finding the square root of 1.44.

2. Group the digits: \( \overline{1}.\overline{44} \).

3. The largest number whose square is less than or equal to 1 is 1. Place 1 as the first digit of the quotient.

4. Subtract 1 from 1, get 0, and bring down 44 to make it 044. Double 1 to get 2 and write it as 2_.

5. The largest digit X to satisfy \( 20X \times X \leq 44 \) is 2, because \( 22 \times 2 = 44 \).

6. Write 2 next to 1 in the quotient, giving 1.2. Since the remainder is 0, the process stops, and the square root of 1.44 is 1.2.

The prime factorization method is a reliable technique for finding the square root of perfect squares. Here’s a step-by-step guide to using this method:

1. Prime Factorization: Start by expressing the number as a product of its prime factors.

   For example, to find the square root of 144:

   144 = 2 × 2 × 2 × 2 × 3 × 3

2. Pair the Prime Factors: Group the identical prime factors into pairs.

   144 = (2 × 2) × (2 × 2) × (3 × 3)

3. Select One Factor from Each Pair: Take one factor from each pair.

   (2 × 2) × (2 × 2) × (3 × 3) = 2 × 2 × 3

4. Multiply the Selected Factors: Multiply the selected factors to get the square root.

   2 × 2 × 3 = 12

   Therefore, the square root of 144 is 12.

Let's look at another example for better understanding:

Example: Finding the Square Root of 3600

1. Prime Factorization: 3600 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5

2. Pair the Prime Factors: (2 × 2) × (2 × 2) × (3 × 3) × (5 × 5)

3. Select One Factor from Each Pair: 2 × 2 × 3 × 5

4. Multiply the Selected Factors: 2 × 2 × 3 × 5 = 60

   Therefore, the square root of 3600 is 60.

Prime factorization helps break down the problem into manageable parts, making it easier to understand and calculate square roots of perfect squares.

• What are the major five steps to find the square root by long division method?

  The five main steps are: divide, multiply, subtract, bring down, and repeat. This systematic approach helps to calculate the square root accurately, even for large numbers.

• How does the long division method work to find the square root?

  The method involves drawing a tableau where the divisor is placed outside and the dividend inside. The quotient is placed above the dividend. The steps involve grouping digits, performing division, multiplying, subtracting, and bringing down the next group, repeating the process until the square root is found.

• How can we find the square root of a decimal number by the division method?

  To find the square root of a decimal number using the division method, you can follow the same steps as for whole numbers. Group the digits in pairs starting from the decimal point, perform the division, multiplication, subtraction, and bring down the next group of digits, including decimal places as needed, and repeat.

• What is the square root of 3249 by the division method?

  The square root of 3249 using the division method is 57.

• What is the square root of 7921 by the division method?

  The square root of 7921 using the division method is 89.

• What is the square root of 2304 by the division method?

  The square root of 2304 using the division method is 48.

• What is the square root of 3250 by the division method?

  The square root of 3250 using the division method is approximately 57.0.

Here are some practice problems along with their solutions to help you master the square root by division method.

1. Find the square root of 441:

   Solution:

   1. Pair the digits: \(\overline{4}\overline{41}\).
   2. Find the largest number whose square is less than or equal to 4. This is 2 (since \(2^2 = 4\)).
   3. Subtract: \(4 - 4 = 0\). Bring down the next pair: \(\overline{41}\).
   4. Double the quotient: \(2 \times 2 = 4\). Write it as 4\_.
   5. Find a digit (x) such that \(4x \times x \leq 41\). This digit is 1 (since \(41 \times 1 = 41\)).
   6. Subtract: \(41 - 41 = 0\).
   7. The quotient is 21. Therefore, \(\sqrt{441} = 21\).

2. Find the square root of 576:

   Solution:

   1. Pair the digits: \(\overline{5}\overline{76}\).
   2. Find the largest number whose square is less than or equal to 5. This is 2 (since \(2^2 = 4\)).
   3. Subtract: \(5 - 4 = 1\). Bring down the next pair: \(\overline{76}\).
   4. Double the quotient: \(2 \times 2 = 4\). Write it as 4\_.
   5. Find a digit (x) such that \(4x \times x \leq 176\). This digit is 4 (since \(44 \times 4 = 176\)).
   6. Subtract: \(176 - 176 = 0\).
   7. The quotient is 24. Therefore, \(\sqrt{576} = 24\).

3. Estimate the square root of 200:

   Solution:

   1. Pair the digits: \(\overline{2}\overline{00}\).
   2. Find the largest number whose square is less than or equal to 2. This is 1 (since \(1^2 = 1\)).
   3. Subtract: \(2 - 1 = 1\). Bring down the next pair: \(\overline{00}\).
   4. Double the quotient: \(1 \times 2 = 2\). Write it as 2\_.
   5. Find a digit (x) such that \(2x \times x \leq 100\). This digit is 4 (since \(24 \times 4 = 96\)).
   6. Subtract: \(100 - 96 = 4\). Bring down the next pair: 00, making it 400.
   7. Double the quotient: \(14 \times 2 = 28\). Write it as 28\_.
   8. Find a digit (y) such that \(28y \times y \leq 400\). This digit is 1 (since \(281 \times 1 = 281\)).
   9. Subtract: \(400 - 281 = 119\).
   10. The quotient is 14.1 (approx). Therefore, \(\sqrt{200} \approx 14.1\).

Here are some helpful visual and video resources to understand the square root by division method more effectively:

• Khan Academy

  Khan Academy offers detailed video explanations of the square root by division method. These videos break down each step and provide visual aids to help understand the concept better. You can watch the video tutorials on their platform:

• BYJU'S

  BYJU'S provides comprehensive video lessons that explain the square root by division method using various examples. These videos are designed to simplify the process and make it easy to follow:

• Cuemath

  Cuemath offers interactive videos and step-by-step tutorials to help students grasp the division method for finding square roots. These resources are ideal for visual learners:

• Toppr

  Toppr features videos that cover various mathematical concepts, including the square root by division method. These videos are created to aid in understanding and provide practice problems: