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Square Root Long Division Method

Introduction to Square Roots

Recognise what a square root is. For example, since 5 × 5 = 25, the square root of 25 is 5.
Understand the difference between non-perfect squares (like 2, 7, 10) and perfect squares (like 4, 9, 16).
Understand the significance of computing square roots in everyday life, science, engineering, and mathematics.

Table of Content:

An Overview of Methods for Square Roots
Understanding the Long Division Method
Basic Concepts of the Long Division Method
Detailed Steps of the Long Division Method
Solved Examples
Practical Examples
Common Mistakes to Avoid
Fun Facts
Conclusion
FAQs on Square Root Long Division Method

An Overview of Methods for Square Roots

Recognise that there are several methods for calculating square roots:

Using logical guesswork as an estimation method
Dividing a number into factors using the prime factorisation method
The long division method is an exact technique that works well with large or decimal numbers.

Find out why the long division method is particularly helpful in situations where high precision is required or calculators are prohibited.

Understanding the Long Division Method

1. Purpose and Importance

Discover how to find precise square roots without a calculator by using the long division method.
Recognise the contexts in which this approach is frequently employed, such as academic assignments or competitive tests.

2. Basic Concepts

putting numbers in pairs, beginning at the decimal point.
gaining knowledge about the long division layout.
Recognising terms related to square roots, such as dividend, divisor, remainder, and quotient.

Detailed Steps of the Long Division Method

Step-by-step process in detail:

1. Grouping Digits

Whole Numbers: Group digits in pairs from right to left.
E.g. 7056 → groups as 70 | 56
Decimals: Group digits in pairs from the decimal point outward.
E.g. 45.678 → groups as 45 | 67 | 80

2. Finding the First Digit

Examine the group on the left.
Determine the biggest number whose square is equal to or less than that group.
For instance, the first digit is 8 if the first group is 70, since 64 (8²) is the closest perfect square.
Put this number at the beginning of your response (the quotient).

3. Subtracting and Bringing Down the Next Pair

From the first group, subtract the first digit's square.
To create a new dividend, bring down the subsequent pair of numbers to the right of the remainder.
Get ready to compute the square root's subsequent digit.

4. Forming the New Divisor

To contribute to the new divisor, double the current quotient.
For the quotient's subsequent digit, leave a blank space adjacent to it.
For example, if your current quotient is 8, double it to get 16. 16- is the initial value of your new divisor..

5. Finding the Next Digit

Trial and error:

To determine which digit provides the best fit without going over the current dividend, insert numbers 0 through 9 into the blank.
Subtract from the current dividend after multiplying the new divisor by the trial digit.

Once found, add that digit to the quotient.

6. Repeating the Process

Keep repeating the steps of:

Doubling the current quotient
Testing digits
Subtracting
Bringing down pairs

Continue until:

All pairs are processed, or
The square root has the desired decimal places of accuracy.

Solved Examples

Example 1

Find √324 using the long division method.

Step 1: Group the digits

Start from right and make pairs:

324→(3) (24)

So, groups are: 3 and 24

Step 2: Find the largest square ≤ first group

First group = 3
The largest square ≤ 3 is 1² = 1
Quotient so far = 1
Subtract:

3−1=2

Bring down the next pair (24):

New dividend = 224

Step 3: Double the quotient

Quotient so far = 1
Double it → 2 × 1 = 2

We now have:

Try digits to fill the blank.

Step 4: Find next digit

Try 8:

→ 28 × 8 = 224

Step 5: Subtract

Subtract 224 from 224:

224−224=0

Remainder = 0.

Thus, the square root of 324 = 18

Answer: √324 = 18

Example 2

Find √2025 using the long division method.

Step 1: Group the digits

2025→(20) (25)

Groups: 20 and 25

Step 2: Find largest square ≤ first group

Largest square ≤ 20 is 16 (4²)

Quotient so far = 4
Subtract:

20−16=4

Bring down next pair → new dividend = 425

Step 3: Double the quotient

Double of 4 = 8

So, we write:

Step 4: Find next digit

Try 5:

→ 85 × 5 = 425

Step 5: Subtract

425−425=0

Thus, √2025 = 45

Answer: √2025 = 45

Example 3

Find √7056 using the long division method.

Step 1: Group the digits

7056→(70) (56)

Groups: 70 and 56

Step 2: Find largest square ≤ 70

Largest square ≤ 70 = 64 (8²)

Quotient so far = 8
Subtract: 70−64=6

Bring down next pair → new dividend = 656

Step 3: Double the quotient

Double of 8 = 16

So:

Step 4: Find next digit

Try 4:

→ 164 × 4 = 656

Step 5: Subtract

656−656=0

Thus, √7056 = 84

Answer: √7056 = 84

Example 4

Find √4489 using the long division method.

Step 1: Group the digits

4489→(44) (89)

Groups: 44 and 89

Step 2: Find largest square ≤ 44

Largest square ≤ 44 = 36 (6²)

Quotient so far = 6
Subtract:

44−36=8

Bring down next pair → new dividend = 889

Step 3: Double the quotient

Double of 6 = 12

So:

Step 4: Find next digit

Try 7:

→ 127 × 7 = 889

Step 5: Subtract

889−889=0

Thus, √4489 = 67

Answer: √4489 = 67

Practical Examples

Example 1: Use the long division method to find the square root of 7056.
Example 2: To two decimal places, find the square root of a non-perfect square, such as 45.678.
To see the procedure in action, see each step detailed in writing.

Common Mistakes to Avoid

failing to properly group digits.
mistakes made when the quotient is doubled.
incorrectly inserting the decimal in the response.
selecting the incorrect number during the trial-and-error process.

Fun Facts

An Old Technique

For square roots, the long division method dates back hundreds of years. Prior to the invention of calculators, mathematicians from ancient India, such as Aryabhata, and later Europe employed comparable methods.

Any Size Number Can Be Used

You can use this method to find the square root of 25 or 25 million. All you need is enough room on your paper and patience.

Precision of Hidden Decimals

This method allows you to find square roots to as many decimal places as you want. You simply continue with the steps, extending pairs of zeros.

Conclusion

Even without a calculator, you can compute square roots with high accuracy if you know how to use the long division method. It creates a solid mathematical foundation that is beneficial for tests, more complex math, and solving problems in everyday life.

Related Links:

Square roots: Ready to master square roots? Try solving a few examples yourself using the long division method and boost your confidence in maths!

Divisibility rule: Make math easy, explore more tricks like divisibility rules on our website today.

Frequently Asked Questions on Square Root Long Division Method

1. How to do long division for square root?

Answer:
To find the square root of a number using the long division method:

Group digits in pairs from right to left (and from decimal point if applicable).
Start with the leftmost pair, find a number whose square is less than or equal to it.
Subtract and bring down the next pair beside the remainder.
Double the quotient and use it as a new divisor's prefix.
Find a digit (X) such that (20 × X + X) × X is less than or equal to the current number.
Repeat the process until you reach the required number of digits or zero remainder.