How to Find Square Root by Long Division Method – Steps
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
1. What is the long division method for square roots?
The long division method for square roots is a step-by-step algorithm to find the square root of any number, particularly useful for non-perfect squares and decimals. It involves pairing digits, finding successive divisors, and performing subtractions to gradually reveal the root.
2. How do I find the square root of 17424 by the division method?
To find the square root of 17424 using the long division method:
1. Pair the digits: (1)(74)(24).
2. Find the largest perfect square ≤ the first pair (1): It's 1 (√1 = 1). This is your first quotient digit.
3. Subtract 1 from 1 (remainder is 0). Bring down the next pair (74).
4. Double the quotient (1 × 2 = 2) and find the next divisor. Then you find a digit (say, x) such that (2x) * x ≤ 74. We find that x = 3 (23 * 3 = 69). So, 3 is your second quotient digit.
5. Subtract 69 from 74 (remainder is 5). Bring down the last pair (24).
6. Double the current quotient (13 × 2 = 26). Find the next digit (y) such that (26y)* y ≤ 524. We find that y = 4 (264 * 4 = 1056). This would exceed 524. If you try y = 2, you will find (262) * 2 = 524. 2 is our third quotient digit.
7. Subtract 524 from 524 (remainder is 0). Hence the square root of 17424 is 132.
3. Where is the square root by division method used in exams?
The square root by division method is frequently tested in various exams, including those focused on arithmetic and calculation skills and in competitive exams where calculators may not be permitted. These exams often assess your understanding of the method and your ability to apply it accurately and efficiently.
4. Can I use this method for decimal square roots?
Yes, the long division method works for decimal square roots. After finding the square root of the whole number part, add pairs of zeros after the decimal point and continue the division process to obtain the desired decimal precision.
5. What is the square root of 42 by the long division method?
Using the long division method, the square root of 42 is approximately 6.48. The exact value is an irrational number; the long division method allows you to find an approximation to any desired level of accuracy.
6. Why do we group digits in pairs from the right in the square root long division method?
Grouping digits in pairs from the right in the square root long division method aligns with the properties of perfect squares and simplifies the algorithm by working systematically from higher place values down. This ensures efficient extraction of the square root.
7. How does the long division method differ from factorization or estimation for square roots?
Factorization is efficient for perfect squares, finding prime factors. Estimation provides approximations. The long division method provides accurate square roots for all numbers (integers and decimals), regardless of whether they are perfect squares.
8. What are common mistakes students make in square root long division?
Common mistakes include:
• Incorrectly pairing digits
• Errors in divisor selection and subtraction
• Mistakes in bringing down digits. Careful, step-by-step calculation minimizes errors.
9. How can I check if my answer is correct without a calculator?
Square your calculated square root. If it closely matches the original number (allowing for rounding errors in decimal roots), your answer is likely correct.
10. What are some tips for increasing speed in square root long division?
Practice is key! Familiarize yourself with perfect squares and learn to quickly estimate suitable divisors during the division process. This improves calculation speed and accuracy over time.
11. Is there a way to use this method to find the cube root of a number?
No, the long division method as described is specifically for finding square roots. Different algorithms are needed for finding cube roots and higher-order roots.
