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. How do you find the square root of a number?
To find the square root of a number, you can use several methods. For perfect squares, prime factorization is easiest. For non-perfect squares, the long division method provides an accurate approximation. Calculators offer the quickest solution for any number. Choose the method that best suits your needs and the complexity of the number.
2. What is the formula for finding the square root?
The square root of a number n is represented as √n or n1/2. This means finding a number that, when multiplied by itself, equals n. For example, √25 = 5 because 5 x 5 = 25.
3. How can I find the square root without a calculator?
For perfect squares, prime factorization is efficient. For non-perfect squares, employ the long division method, which involves a step-by-step process of estimation and subtraction to arrive at an approximate answer. Both methods allow accurate calculation without a calculator. Practice increases speed and accuracy.
4. What is the square root of 225?
The square root of 225 is 15, because 15 multiplied by itself (15 x 15) equals 225.
5. How do you find the square root of a decimal or fraction?
Finding the square root of a decimal or fraction can involve converting it to an equivalent form, if possible, to make calculations easier. For example, converting a fraction to a perfect square simplifies the process. Then, use prime factorization or long division. A calculator is also a convenient tool for direct calculation.
6. What are perfect squares?
A perfect square is a number that can be obtained by squaring an integer (a whole number). For instance, 9 is a perfect square because it is 3 squared (3 x 3 = 9). Other examples include 1, 4, 16, 25, and so on. Identifying perfect squares simplifies root calculation.
7. What is the square root equation?
A square root equation involves a radical symbol (√). To solve it, isolate the radical term first. Then, square both sides of the equation to eliminate the radical. Simplify the resulting equation to find the solution (or solutions). Always check your answer(s) to ensure they work in the original equation.
8. What if the number is negative? Can I still find the square root?
Real numbers do not have square roots if they are negative. The square roots of negative numbers are imaginary numbers, involving the imaginary unit 'i' where i2 = -1. These are studied in more advanced mathematics.
9. How to quickly check if a number is a perfect square?
One way to check if a number is a perfect square is through prime factorization. If each prime factor appears an even number of times, the number is a perfect square. Alternatively, you can try to find an integer whose square equals your number. If you find one, it's a perfect square.
10. Are there mental math tricks for finding square roots?
Yes, there are several techniques. One common approach is to identify the closest perfect squares and then use estimation to refine your guess. For example, to find √75, recognise that 8 x 8 = 64 and 9 x 9 = 81. Therefore, √75 is between 8 and 9, closer to 9.
11. What are some real-world applications of square roots?
Square roots are essential in various fields. In geometry, they’re used to calculate distances (Pythagorean theorem), areas (circles, triangles), and volumes (spheres, cones). They also appear in physics (kinematics, energy calculations), finance (compound interest), and many other areas.
12. How accurate are the methods for finding square roots?
The accuracy depends on the method used. For perfect squares, prime factorization yields an exact answer. The long division method gives an approximation, with accuracy increasing as you carry out more steps. Calculators offer high precision, limited only by the calculator's display capabilities and internal algorithms.
