How to Find Square Root Using Long Division Method with Formula and Solved Examples
The concept of square root long division method plays a key role in mathematics and is widely used for finding square roots of large or non-perfect square numbers without using calculators. Knowing this method helps students in exams and real-life calculations.
What Is Square Root Long Division Method?
The square root long division method is a step-by-step procedure to find the square root of any number, especially non-perfect squares. It involves pairing digits, finding divisors, and using subtraction and division—similar to regular long division. You’ll see this used in topics such as square roots, decimal expansions, and competitive exam problem-solving.
Key Formula for Square Root Long Division Method
Here’s the standard formula: \( \sqrt{N} \) where N is the number. The process involves:
Group the digits of N in pairs (from right to left), then repeatedly:
- Find a divisor where (divisor × quotient) ≤ current number
- Subtract, bring down next pair, double the quotient, and repeat
Cross-Disciplinary Usage
Square root long division method is not only useful in Maths but also supports Physics (speed/area calculations), Computer Science (algorithmic accuracy), and everyday logical reasoning. Students preparing for JEE, Olympiad, and board exams often need this method for precision when calculators are not allowed.
Step-by-Step Illustration
Let’s find the square root of 361 using the long division method:
1. Pair the digits from right: \(\bar{3}\)\(\bar{61}\)2. Find the largest square ≤ 3: 1 × 1 = 1.
3. Subtract 1 from 3: remainder 2.
4. Bring down the next pair (61), making 261.
5. Double the first quotient (1): 2. Find a digit x, so (2x)×x ≤ 261.
6. Try x = 9: (20+9)×9 = 29×9 = 261.
7. Subtract 261 – 261 = 0.
8. The quotient (19) is the square root of 361.
Square Root Long Division Table for 361
| Step | Digits/Pair | Divisor/Quotient | Remainder |
|---|---|---|---|
| 1 | 3 | 1 | 2 |
| 2 | 61 (brought down) | 2⎯_9 (try x=9, divisor 29) | 0 |
Speed Trick or Vedic Shortcut
For perfect squares, quickly recall squares from 1 to 25 (see: Square Root from 1 to 25) to estimate. For decimals, after finishing the whole-number part, add pairs of zeroes and continue division for decimal places.
Example Trick: For numbers ending in 6, try 4 or 6 as the last digit of root, as squares ending in 6 are from 4 or 6.
Try These Yourself
- Find the square root of 55225 by the long division method.
- Calculate √68 by long division.
- Find √42 up to two decimal places using long division.
- What’s the square root of 17.64 by this method?
Frequent Errors and Misunderstandings
- Forgetting to pair digits correctly (especially for decimals).
- Using wrong divisor expansion (not doubling the current quotient).
- Not bringing down digit pairs together.
- Putting the decimal in the wrong spot.
Relation to Other Concepts
The square root long division method is closely connected to perfect squares, square root estimation, and algorithms behind square root calculators. Mastering this boosts your confidence for higher number root-finding too.
Classroom Tip
To remember the square root by long division method easily: always “pair-digits, double-quotient, repeat.” Teachers at Vedantu use flow charts and step tables to make the process visual—helpful for both Class 7 and Class 8 students.
We explored square root long division method—definition, steps, solved examples, shortcuts, and common issues. For more practice, try Vedantu’s square root questions or use the Find Square Root tool. Learning this method makes exams and logical reasoning much easier!
More learning helps: Square Root Table, Estimating Square Root, Square Root Finder
FAQs on Square Root Long Division Method Explained Step by Step
1. What is the square root long division method?
The square root long division method is a systematic technique used to find the square root of large or non-perfect square numbers by division. It works similarly to long division and is especially useful when the square root is not obvious.
- Digits are grouped in pairs from right to left.
- The largest possible square is subtracted at each step.
- The process continues to get decimal places if required.
2. How do you find a square root using the long division method?
To find a square root using the long division method, follow a step-by-step pairing and subtraction process.
- Step 1: Group digits in pairs from right to left.
- Step 2: Find the largest number whose square is ≤ the first pair.
- Step 3: Subtract and bring down the next pair.
- Step 4: Double the quotient and find a suitable digit to multiply.
- Step 5: Repeat until the required accuracy is reached.
3. Can you give an example of the square root long division method?
Yes, the square root of 529 using the long division method is 23.
- Group digits: 5 | 29
- Largest square ≤ 5 is 2² = 4 → remainder 1
- Bring down 29 → 129
- Double 2 → 4_ ; find digit 3 so that 43 × 3 = 129
- Remainder = 0
4. Why do we group digits in pairs in the square root long division method?
Digits are grouped in pairs because each step of the square root process works with powers of 100 (since 10² = 100).
- Pairing helps manage place values correctly.
- Each pair corresponds to one digit in the square root.
- This ensures accurate placement of digits in the quotient.
5. How do you find the square root of a non-perfect square using long division?
To find the square root of a non-perfect square, continue the long division method by adding decimal pairs of zeros.
- Place a decimal point in the quotient.
- Add pairs of 00 after the decimal.
- Continue the same doubling and subtracting process.
6. What is the formula used in the square root long division method?
The key working formula in the square root long division method is based on (20a + b) × b.
- ‘a’ represents the current quotient.
- Double the quotient (2a).
- Form a new divisor as (20a + b).
- Multiply by b and subtract.
7. What is the difference between the square root long division method and the prime factorization method?
The long division method finds square roots through repeated division steps, while prime factorization uses factor pairs.
- Long division works for both perfect and non-perfect squares.
- Prime factorization is mainly useful for perfect squares.
- Long division can give decimal answers.
8. How do you check your answer in the square root long division method?
You can check your answer by squaring the obtained result and verifying it equals the original number.
- If √625 = 25, then check 25² = 625.
- For decimals, multiply the result by itself.
9. What are common mistakes in the square root long division method?
Common mistakes in the square root long division method include incorrect pairing and wrong subtraction steps.
- Not grouping digits correctly in pairs.
- Forgetting to double the quotient.
- Choosing a digit that makes the product too large.
- Misplacing the decimal point.
10. When should you use the square root long division method?
The square root long division method should be used when calculating square roots of large numbers or non-perfect squares without a calculator.
- Useful in exams where calculators are not allowed.
- Helps find square roots to several decimal places.
- Provides a clear manual computation method.
