Square Root Calculator
√a

The common way or formula for calculating the square root

The square root of a number is the value that, when multiplied by itself, gives the original number. Mathematically, it’s represented as:

√x = y where y x y = X

While this is simple for perfect squares, non-perfect squares and decimals require more advanced methods.

Methods to Calculate Square Roots

Calculating square roots can be approached in multiple ways, depending on the type of number and the level of precision required. From manual techniques like the long division method to quick estimation tricks and advanced tools like scientific calculators, each method has its unique applications. Below, we explore the most common and effective ways to calculate square roots, catering to both students and professionals.

1. By Long Division Method

The long division method is a systematic way to calculate square roots, especially for large numbers. Here's how:

Calculate the square root of 529 using the long division method.

• Divide the digits into pairs from right to left: 5 29.
• Start with the leftmost pair, 5. The largest number whose square is ≤ 5 is 2 (2²=4)
• Subtract 4 from 5:5−4 = 1. Bring down the next pair, 29, making it 129.
• Double the divisor (2×2 = 4) and find the next digit x such that 40x × x ≤129. Here, x= 3 (43×3 = 129)
• Subtract 129 −129 = 0
• Thus, the square root of 529 =23

2. Using Tricks and Estimation

For quick approximations, use these tricks:

• Find the nearest perfect squares and estimate the value in between.
• Use fractions for closer approximations, especially for non-perfect squares.

Find the approximate square root of 50.

Steps:


1. The nearest perfect squares are 49 (7²) and 64 (8²).
2. 50 lies closer to 49. Start with 7 as the estimate.
3. For more precision, use the fraction: 7+ 50 - 49 = 7 + 1 ≈ 7.07
4. 2 x 7 14

Thus, √50 ≈ 7.07

3. For Decimal Numbers

• Multiply the decimal by powers of 10 to convert it into a whole number.
• Find the square root of the resulting number.
• Divide the answer by the same power of 10 used earlier.

Find the square root of 0.25.

Steps:


1. Multiply 0.25 by 100 to make it a whole number: 0.25 × 100 = 25.
Find the square root of 25: √25 = 5. Divide the result by 10: 5 ÷ 10 = 0.5.

Thus, √0.25 = 0.5

4. For Prime Numbers and Non-Perfect Squares

Prime numbers and mperfect square roots require more precision:

• Use the long division method or estimation.
• Alternatively, apply Newton’s Approximation Formula for faster results.

Find the square root of 19 (a prime number).

Using Estimation:

1. The nearest perfect squares are 16 (4²) and 25 (5²).
2. 19 lies closer to 16. Start with 4 as the estimate.
3. Refine using Newton’s Approximation: 

x=4, x new = (x + 19 / X) / 2

Substituting x = 4:

X new = (4 + 19 / 4) / 2 = (4 + 4.75) / 2 = 4.375

Thus, √19 ≈ 4.375

5. Using Scientific Calculators

• Input the number and press the square root (√) function.
• Adjust the decimal precision as needed for accuracy.

How to Solve Complex Square Root Problems

Square root problems can involve decimals, non-perfect squares, and prime numbers. Here’s how to approach them:

1. For Decimals

• Simplify the decimal by converting it into a fraction.
• Calculate the square root of the numerator and denominator separately.

2. For Non-Perfect Squares

• Use estimation methods to find approximate values.
• Apply iterative methods for precise results.

3. For Prime Numbers

• Prime numbers often result in irrational square roots. Use the long division method for better accuracy.

4. General Tips

• Break complex problems into smaller steps.
• Practice using both manual methods and calculators to build confidence.

5. Why Learn About Square Roots?

Square roots are a cornerstone of the CBSE/ICSE curriculum, appearing in chapters like:

• Number Systems
• Algebra
• Mensuration

They are also essential in practical applications:

• Physics: Calculating forces and distances.
• Geometry: Finding areas of shapes like circles.
• Design and Architecture: Determining dimensions and layouts.

How to solve square root problems without a calculator.

While calculators are helpful, manual methods enhance problem-solving skills. Here's how to tackle square roots manually:

• For Perfect Squares: Memorize common values (e.g., √4=2).
• For Non-Perfect Squares: Use the long division method or tricks.
• For Decimals: Simplify the number into a fraction first.
• For Prime Numbers: Apply estimation or iterative approaches.