Square Root Formula

A square root is common function in mathematics. A square root is widely used in different applications in different fields of mathematics and physics. Here is a guide to finding the square root of a number by square root formula.

Square Root Formula

What is a Square root?

The square root of the number y whose square is x. The square root is denoted by \(\sqrt{}\)

We find the square root of a number by the following methods:

i) By Prime Factorisation

ii) By Long Division

iii) By Repeated subtraction method

1.By Prime Factorisation: Steps to find the square root of a perfect square by using the prime factorization method.
Step I: Obtain the given number.
Step II: Reduce the given number into prime factors by successive division.
Step III: Now make pairs of prime factors in such a way that both the factors in each pair are equal.

Step IV: Take one factor from each pair and find the product of these factors.
Step V: The product obtained by multiplying the factors is the required square root.

2. By Long Division: Steps to find the square root of a number by long division method.

Step 1: Firstly, we place a bar on every pair of digits starting from the unit digit. If the number of digits in it is odd, we put a bar on the single-digit too. For example, we take 729. So 1^st bar is on 29 and 2^nd bar is on 7

Step 2: Now we find the largest number whose square is less than or equal to the 1^st number.
\((2^{2} < 7< 3^{2})\). We take 2 and divide and get the remainder = 3.

Step 3: Now we bring down the next bar number i.e.29. So, the new dividend is 329.

Step 4: For new divisor, we add the divisor 2 and quotient 2 that gives us 4.

Step 5: Number taken is the product of a new divisor and this digit is equal to or less than 329 (new dividend).
In this case,

47 × 7 = 329.

The new digit is 7. We get remainder as 0.
∴ \(\sqrt{729}\) = 27.

3. Repeated subtraction method: In this method, the given number is subtracted by 1, 3, 5, 7,… at every step till you get zero at the end. The number of steps in the solution is the required square root.

Solved Examples

Q1. In a concert hall, the number of rows is equal to the number of chairs in each row. If the capacity of the concert hall is 2025, find the number of chairs in each row.

Solution: Let the number of chairs in each row in the concert hall be x.

Then, the number of rows = x.

Total number of chairs in the concert hall = \((x \times x) = x^{2}\)

But, the capacity of the concert hall = 2025

Therefore, x² = 2025 = (5 × 5 × 3 × 3 × 3\)

x = \((5 \times 3 \times 3)\) = 45

Number of rows in the concert hall is 45.

Q 2: Find the square root of 49.

Solution : (i) 49 -1 = 48
(ii) 48 – 3= 45
(iii) 45 – 5 = 40
(iv) 40 – 7= 33
(v) 33 – 9 = 24
(vi) 24 – 11 = 13
(vii) 13 – 13 = 0
Here, the total number of subtractions is 7.
∴ \(\sqrt{49} = 7\)