Square Root

Introduction

The square root is actually applied to square numbers to find their initial value, from which they were squared. The square root acts as a base for entire mathematics. It is an important topic that needs to be learned properly. In this article we will be learning, what square root is and different methods to find the square root. It is necessary to have a clear understanding of this concept to solve higher-level problems later. 

Square root

The square root of a number is a value, which is multiplied by itself to give the initial number. It can also be said as the ½ power of the number. The inverse of squaring a number is the square root. 

Let us assume a to be the square root of b, then it can be written as a=√b. Here b is the square of a, which is b = a2. The sign ‘√’ is used to represent the square root of any number. The square root of any squared number is the original number. 

How to find square root:

The square root of any value can be found by the prime factorization method, long division method, subtraction method, and estimation method. In this article, we learn how each of these methods works. 

Square root by prime factorization method:

This method is preferred to find the square root of any perfect square number. Steps involved in finding the square root by prime factorization method:

• The given number is written in the form of its prime factors.
• Prime factors of similar numbers are written in pairs.
• One factor is chosen from each pair.
• The product of chosen factors is the square root of the given number.

Example: Find the square root of 1225.

1225 = 5 x 5 x 7 x 7. 

   From each pair, one factor is chosen and multiplied.

   Multiplying the pairs,

          5 x 7 = 35.

      ∴√1223 = 35.

Square root for non-perfect number by prime factorization method:

• The given number is written in the form of its prime factors.
• Prime factors of similar numbers are written in pairs.
• One factor is chosen from each pair and multiplied, numbers without pair remain inside the square root. 

      √6125 = √ 7 x 7 x 5 x 5 x 5.

      √6125 = 35√5. 

Square root by subtraction method:

This method can be used to find the square root of a perfect square number. 

Steps involved in finding the square root of a given number:

• The given number is subtracted by consecutive odd integers.
• The number is subtracted till the endpoint is zero.
• A number of times subtracted is the square root of the given value.

Find square root of 36,

36 – 1 = 35

35 – 3 = 32

32 – 5 = 27

27 – 7 = 20

20 – 9 = 11

11 – 11 = 0

   The given number is subtracted 6 times, hence the square of 36 is 6.

Square root by long division method:

This method can be used to find the square root of any given number. It is easier to use and gives more accurate results. Let us take an example and learn along with that. 

Let us take 441 and place a bar to a pair of numbers. Place bar starting from unit’s place. 41 is barred together. If the given number is odd, there will be a number left out without pair 4 (41) ̅.

The dividend should be chosen in such a way that the square of the number should be less than or equal to the left extreme pair of as a given number. So, 2 is chosen as a dividend. Here 2 is the quotient and 0 is the remainder. 

Now the first pair right next to the first number is brought down. And the quotient and dividend are added and written as new dividends and a number must be filled in its unit place. 

Now the dividend is 21 and the quotient 21 is the root.

Estimation method: 

The estimation method can be used to approximately calculate the values of the square root of a number. The square root of 9 is 3 and 16 is 4. To find the square root of 11, the value must lie between 3 and 4.

Let us pick random numbers, 

3.32 = 10.89

3.52 = 12.25

The value 3.3 is closer to the given number. So square root of 11 is approximately 3.3.

Conclusion

The square root is an important operation in maths, just like any other arithmetic operation. It is necessary to learn how to solve a problem. It is important to solve problems with roots, to further advance into higher-level maths.