What is Square Root?

Define a Square Root

A square root is a systemic function applied in the field of Mathematics. It is described as the value of a natural number written as y = \[\sqrt{a}\] . Here, the square root of ‘a’ is equal to ‘y’, where ‘a’ is a natural number. It can also be expressed as y\[^{2}\] = a. Therefore, the square root is defined as a number multiplied by itself would produce the original number. For example, 4 × 4 =16, and therefore the square root of 16 is 4. Square roots have been used since ancient times and provide a very simplistic way to solve numerical problems in algebra and geometry.


The Symbol of Square Root

The square root symbol is unique and well-known. It is represented as ‘\[\sqrt{}\]’ The symbol for a square root is also known as a radical and the number present under the root symbol is known as radicand. 

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How to Find the Square Root?

To find the square root of a number, dependents on the type of data or number provided. 

In the case of perfect square numbers, it is simple to obtain the original number by multiplying the prime number twice.

An example of the square root of \[\sqrt{4}\], we know that 2 × 2 = 4.

In the case of imperfect squares, a fraction is obtained instead. This may be a complicated process to follow, but, with a little practice, one can go a long way to solve problems related to this.

An example of the square root of 2\[\sqrt{2}\] = 1.414

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Types of Square Root Methods


A. Square Root by Prime Factorization

The prime factorization method to find the square root of a number is easy. Firstly, we factorization the number under the root sign followed by pairing them in two.

For example,

the square root of 9 is \[\sqrt{9}\] = \[\sqrt{3 \times 3}\] = 3 

An easy way to remember how square roots work is that the inverse of squaring a number is finding its root. For example,

Hence, 12 = 1, the square root of 1 is 1

Hence, 42 = 16, square root of 16 is 4, and so on

When we take the prime factors of a number and its square, 

For example 12 and 144,

12 = 2 х 2 х 3

144 = 2 х 2 х 3 х 2 х 2 х 3 

Prime factorization of a square number provides prime factors that occur two times that of the number itself. 

 

B. Square Root By Long Division

Below is an example, given to understand how long division works with the square root.

1. Let us take 484 as the number whose square root is to be evaluated. A bar is placed over the pair of numbers starting from the unit place. If there is a number, then a bar is placed over the left digit as well. 

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2. The divisor is the largest number whose square is less than or equal to the number on the extreme left of the number. On dividing, write the quotient. 

3. Here the quotient is 2 and the remainder is 0.

4. Next, the number with the bar is brought down to the right side of the remainder. Here, in this case, we bring down 84. Now, 84 is our new dividend.

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5. The value of the quotient is doubled and entered in the blank space on the right side.

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6. Select the largest digit for the unit place of the divisor (4_) such that the new number, when multiplied by the new digit at unit’s place, is equal to or less than the dividend (84).

In this case,  42 × 2 = 84. So the new digit is 2.

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7. When the remainder is 0 there is no divisor left to divide, therefore, \[\sqrt{484}\] = 22

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FAQs (Frequently Asked Questions)

1. What are Some of the Applications of a Square Root?

The square root is a systemic function that is used to solve complex equations and numerical problems. Square roots are a big asset in all the areas of algebra and trigonometry. The formula is devised from square roots that help to solve algebraic equations. For example, the area of a square is the product of its sides where all the sides are of equal length. Therefore, if one of the sides of the square is known, the area of the square can be calculated easily.

 

Using the formula, Area of square = Side х Side = Side²  

 

The square root of the area can further be used to calculate the length of each side of the square. This applies to all definite shapes. 

2. Give a List of Common Square Roots Devised for Whole Numbers.

Below, a table exhibits a list of square root values for whole numbers starting from 1 to 29. By remembering the values for these whole numbers, one can master the art of solving complex equations and simplify them. 

 

Number

Square Root(√)

Number

Square Root(√)

1

1

18

4.243

2

1.414

19

4.359

3

1.732

20

4.472

4

2.000

21

4.583

5

2.236

22

4.690

6

2.449

23

4.796

7

2.646

24

4.899

8

2.828

25

5

9

3

26

5.099

10

3.162

27

5.196

11

3.317

28

5.292

12

3.464

29

5.385

13

3.606

 

 

14

3.742

 

 

15

3.873

 

 

16

4

 

 

17

4.123