How to Find Square Root by Long Division Method (with Example)
The concept of find square root plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you are studying for school exams, entrance tests, or simply want to solve practical problems quickly, mastering how to find square root can give you a clear advantage. Vedantu’s Maths guidance helps build a strong foundation in this core topic, using both step-by-step solutions and calculator tricks.
What Is Find Square Root?
A square root is defined as a value that, when multiplied by itself, returns the original number. For example, the square root of 25 is 5, because 5 × 5 = 25. You’ll find this concept applied in areas such as estimating lengths, solving algebraic equations, and geometry problems. Knowing how to find the square root is also crucial when working with squares and square roots, Pythagoras’ theorem, and in daily measurements.
Key Formula for Find Square Root
Here’s the standard formula: \( \sqrt{n} = x \), where \( x^2 = n \). In exponent form, this can be written as \( n^{\frac{1}{2}} \).
Cross-Disciplinary Usage
Find square root is not only useful in Maths but also plays an important role in Physics (e.g., velocity and distance calculations), Computer Science (e.g., algorithms), and logical reasoning. Students preparing for JEE, NEET, or Olympiad exams will see its relevance in many high-scoring questions.
Step-by-Step Illustration
- Let’s find the square root of 81 by the prime factorization method.
1. Break 81 into prime factors: 81 = 3 × 3 × 3 × 3.
2. Make pairs: (3 × 3) and (3 × 3).
3. Take one number from each pair: 3 × 3 = 9.
4. So, √81 = 9. - Now, try the long-division method for a non-perfect square, like 50.
1. Pair the digits of 50 from right: '50'.
2. Find the largest number whose square is ≤ 50 (it is 7, since 7×7=49).
3. Subtract 49 from 50 (50-49=1), bring down two zeros to make 100.
4. Double the number above (7×2=14), guess the next digit (let’s try 0, then 1, then 7).
5. Continue the process for more decimal places if needed.
6. Approximate √50 ≈ 7.07.
Speed Trick or Vedic Shortcut
Here’s a quick shortcut that helps solve problems faster when working with find square root. Many students use this trick during timed exams to save crucial seconds.
Example Trick: If you want the square root of a number between two perfect squares (like 45, between 36 and 49):
- Find the nearest perfect squares (36 = 6², 49 = 7²).
- Check how far 45 is from 36 (it’s 9), and from 49 (it’s 4).
- Since 45 is closer to 49, estimate the square root: Try somewhere between 6.5 and 6.8.
- Try 6.7: 6.7 × 6.7 = 44.89. So, √45 ≈ 6.7.
Shortcuts like estimation and using squares of digits save time in MCQs and Olympiad questions. Vedantu’s live online tutorials reveal many such tricks for fast calculations.
Try These Yourself
- Find the square root of 121.
- Is 144 a perfect square? What is its square root?
- Find square root of 2 correct to two decimal places.
- List all perfect squares between 100 and 200.
- Find square root of 16/81.
Frequent Errors and Misunderstandings
- Assuming find square root is the same as just dividing the number by 2.
- Pairing factors incorrectly when using the prime factorization method.
- Missing decimal values or digits in the long division method.
- Forgetting that negative numbers do not have real square roots (they become imaginary instead).
Relation to Other Concepts
The idea of find square root connects closely with topics such as square numbers and prime factorization. Mastering this helps with understanding more advanced concepts like quadratic equations, simplification of surds, and coordinate geometry.
Classroom Tip
A quick way to remember find square root: if the unit digit of a number is 2, 3, 7, or 8, it is never a perfect square. Also, always check if the number of zeros in a number is even to decide if its square root will be a whole number. Vedantu’s teachers use these visual cues and fun mental tricks in live sessions for faster learning.
Square Root Table (1 to 20)
| Number | Square Root | Perfect Square? |
|---|---|---|
| 1 | 1 | Yes |
| 2 | 1.414 | No |
| 3 | 1.732 | No |
| 4 | 2 | Yes |
| 5 | 2.236 | No |
| 9 | 3 | Yes |
| 16 | 4 | Yes |
| 20 | 4.472 | No |
Wrapping It All Up
We explored find square root—from definition, formula, speed tricks, typical mistakes, and links to related topics. Practice these methods, and use Vedantu’s online square root calculator for instant answers or concept revision before exams. With a little practice, you will work out square roots confidently and accurately every time!
Handy Internal Links for More Practice
- Square Root Calculator – Get instant answers with steps shown.
- Square Root Table – Perfect for last-minute revision and MCQs.
- Estimating Square Root – Master quick mental approximations.
- Square Numbers for Kids – Get comfortable with perfect squares and their roots.
FAQs on How to Find Square Root – Stepwise Methods & Examples
1. What is the definition of a square root and what symbol is used to represent it?
A square root of a number is a specific value that, when multiplied by itself, results in the original number. For example, the square root of 49 is 7 because 7 × 7 = 49. It is primarily represented by the radical symbol (√). Therefore, the square root of a number 'a' is written as √a. It can also be expressed using a fractional exponent as a1/2.
2. What are the two main methods for finding a square root without using a calculator?
The two primary methods taught in the CBSE/NCERT syllabus for manually calculating a square root are:
- Prime Factorisation Method: This method is ideal for numbers that are perfect squares. It involves breaking the number down into its prime factors and creating pairs of identical factors.
- Long Division Method: This is a more versatile method that works for any number, including non-perfect squares and decimals, allowing you to find an approximate value to any desired precision.
3. How can you find the square root of a perfect square like 324 using an example?
To find √324 using the prime factorisation method, follow these steps:
- Step 1: Decompose 324 into its prime factors.
324 = 2 × 2 × 3 × 3 × 3 × 3. - Step 2: Group the factors into identical pairs.
(2 × 2) × (3 × 3) × (3 × 3). - Step 3: From each pair, take one factor and multiply them together.
2 × 3 × 3 = 18.
Thus, the square root of 324 is 18.
4. What is the step-by-step process to find the square root of a non-perfect square, for example, 1069, using long division?
The long division method for finding √1069 involves these steps:
- Step 1: Pair the digits from the right side: 10 69.
- Step 2: Find the largest number whose square is less than or equal to the first pair (10). This is 3 (as 3²=9). Write 3 as both the divisor and the quotient. Subtract 9 from 10, leaving a remainder of 1.
- Step 3: Bring down the next pair of digits (69) to form the new dividend, 169.
- Step 4: Double the current quotient (3) to get 6. This becomes the tens digit of the new divisor. Find a digit 'x' such that 6x × x ≤ 169. Here, x=2 works (62 × 2 = 124).
- Step 5: Place 2 in the quotient (making it 32) and subtract 124 from 169. The remainder is 45.
This shows that 1069 is not a perfect square, and its square root is approximately 32.7, lying between 32 and 33.
5. What is the correct way to find the square root of a fraction or a decimal?
The approach depends on the type of number:
- For Fractions: The rule is √(a/b) = √a / √b. You find the square root of the numerator and the denominator separately. For example, √(16/25) = √16 / √25 = 4/5.
- For Decimals: The long division method is the most reliable. You must pair the digits starting from the decimal point—pairing the whole number part from right to left and the decimal part from left to right. The decimal point in the quotient is placed as soon as the first pair after the decimal is brought down.
6. What is the conceptual difference between the square root of a perfect square and a non-perfect square?
The key difference lies in the nature of the result. A perfect square (like 36) has a square root that is a rational number (a whole number or a terminating decimal, e.g., 6). A non-perfect square (like 37) has a square root that is an irrational number—a non-terminating, non-repeating decimal. Therefore, for non-perfect squares, we can only find an approximate value, not an exact finite answer.
7. Why does the radical symbol (√) give only a positive answer, but an equation like x² = 25 has two answers?
This addresses a common misconception. The radical symbol (√) specifically denotes the principal or positive square root. By convention, √25 is defined to be only +5. However, when solving an algebraic equation like x² = 25, you are asked to find all possible values of 'x' that satisfy the equation. Since both (+5)² = 25 and (–5)² = 25, the equation has two solutions: x = 5 and x = -5.
8. Why is it impossible to find the square root of a negative number in the real number system?
In the real number system, squaring any number—whether positive or negative—always results in a non-negative value (e.g., 4 × 4 = 16 and -4 × -4 = 16). There is no real number that, when multiplied by itself, can produce a negative result. The concept of square roots for negative numbers, like √-1, is defined outside the real number system and belongs to the set of imaginary numbers, a topic studied in higher mathematics.
9. How does the concept of finding a square root relate to exponents?
Finding a square root is the inverse operation of squaring a number (raising it to the power of 2). This relationship is directly represented in the laws of exponents. Taking the square root of a number 'x' is mathematically identical to raising 'x' to the power of 1/2. For instance, √9 can be written as 91/2, which equals 3, just as 3² = 9.
10. Where is finding a square root important in real-world subjects like geometry and physics?
Square roots are fundamental in many practical fields:
- In Geometry: The Pythagorean theorem (a² + b² = c²) uses square roots extensively to find the length of a side of a right-angled triangle. It's also used to find the radius of a circle from its area (A = πr²).
- In Physics: Many formulas for calculating distance, time, and velocity involve square roots. For example, the time it takes for an object to fall a certain distance under gravity is calculated using a square root.
- In Construction: Engineers and architects use square roots to calculate dimensions, surface areas, and ensure structural stability.
