 # Square Root

The Square root of a number is the value which when multiplied by itself gives the original number. For example, the square root of 4 is 2, because when we multiply 2 by itself, it gives the result as 4 only. If a is a number, then the square root of a is represented as √a.

The square root is an important function of principal mathematics, that was found and developed a long time ago. Its history originates from all across the world, spanning from Ancient Greece to Ancient India.  In this article, let us discuss in detail about the definition, formula and get the values of square roots from 1 to 25  with some shortcut tricks.

## Definition

The square root of any number gives the same number when multiplied by itself.

For example, √(m.m) = √(m2) = m

Also, The square root function of a number that helps to map a set of non-negative real numbers onto itself. It is usually represented as:

f(x) = √x

For example, the square root of 2 is denoted as √2. To gain a better understanding of square roots, let us take another example as this will show us exactly how to find a square root of a number.

Consider a number ‘a’ acting as the square root of a number ‘y’ such that y2 = a, where multiplying the number y with itself will lead to its square root (y.y).

## Square Root Symbol

The square root symbol is also called as a radical symbol and is usually represented as ‘. To represent a number ‘x’ as a square root using this symbol can be written as:

√xwhere x is the number itself. The number under the radical symbol is called the radicand. For example, the square root of 6 is also represented as radical of 6. Both represent the same value.

## Square Root Formula

Using the above explanation, we can conclude that,

y.y = y2 = a; where ‘a’ is the square of a number ‘y’ or we can say y is equal to square root of a,i.e., y = √a

## How to Solve Square Root Equation

To solve the square root equation we need to follow the below steps:
Isolate the square to one of the sides (L.H.S or R.H.S)
Square both the sides of the given equation
Now solve the rest equation

Let us understand the steps with examples.
Example: Solve √(4a+9) – 5 = 0
Solution: Given,
√(4a+9) – 5 = 0
Isolate the square root term first.
√(4a+9) = 5
Now on squaring both the sides, we get;
4a+9 = 52
4a + 9 = 25
4a = 16
a = 16/4
a = 4

## Square Roots Table

Here is the list of the square root of numbers from 1 to 50.

 √1 1 √18 4.2426 √35 5.9161 √2 1.4142 √19 4.3589 √36 6 √3 1.7321 √20 4.4721 √37 6.0828 √4 2 √21 4.5826 √38 6.1644 √5 2.2361 √22 4.6904 √39 6.2450 √6 2.4495 √23 4.7958 √40 6.3246 √7 2.6458 √24 4.899 √41 6.4031 √8 2.8284 √25 5 √42 6.4807 √9 3 √26 5.099 √43 6.5574 √10 3.1623 √27 5.1962 √44 6.6332 √11 3.3166 √28 5.2915 √45 6.7082 √12 3.4641 √29 5.3852 √46 6.7823 √13 3.6056 √30 5.4772 √47 6.8557 √14 3.7417 √31 5.5678 √48 6.9282 √15 3.873 √32 5.6569 √49 7 √16 4 √33 5.7446 √50 7.0711 √17 4.1231 √34 5.831

## How do you Figure out Square Roots?

Many mathematical operations have an inverse operation such as division is the inverse of multiplication, subtraction is the inverse of addition. Similarly, Squaring has an inverse operation called square root.  Finding the square root of a number is the inverse operation of squaring that number.

Example: Square of 7 = 7 x 7 = 72 = 49

The square root of 49, 49 = 7

But for some values such as square root of 10, 13, 17, we cannot use this method, as these numbers are not perfect square numbers.

## How to Find Square Roots Without Calculator?

This is quite an interesting way to figure out the square root of a given number. The procedure completely based on the method called “guess and check”

Guess your answer, and verify. Repeat the procedure until you have the desired accurate result. Mostly used to find the square roots of numbers that aren’t perfect squares. We can also use the long division method to find the square root of a number.

## Squares and Square Root

Let us see the value of squares and the square root of squares of the same numbers here.

 Numbers Squares Square root 0 02 = 0 √0 = 0 1 12 = 1 √1 = 1 2 22 = 4 √4 = 2 3 32 = 9 √9 = 3 4 42 = 16 √16 = 4 5 52 = 25 √25 = 5 6 62 = 36 √36 = 6 7 72 = 49 √49 = 7 8 82 = 64 √64 = 8 9 92 = 81 √81 = 9 10 102 = 100 √100 = 10

## Applications of Square Roots

The square root formula is an important section of mathematics that deals with many practical applications of mathematics and it also has its applications in other fields such as computing. We can use the square root values while simplifying the problem. For example, if we are having the equation in the radical form such as the square root of 5(√5), it is quite difficult to solve the equation. So, the substitution of the square root value helps to simplify the equation. Computation can simply be done using our personal calculators that have the square root function within them to get the square root of any number.

It also has a whole list of other functions available such as in geometry, where one can easily map the area of a square to its side length. It also has a whole host of other interrelated mathematical functions and uses; such as its applications in finding out formulas for roots of quadratic integers, quadratic fields and quadratic equations.

## Square Roots Examples

Let us understand this concept with the help of an example:

Example 1:

Solve 10 to 2 decimal places.

Solution:

Step 1:  Select any two perfect square roots that you feel your number may fall in between.

22 = 4; 32 = 9, 42 = 16 and 52 = 25

Choose 3 and 4 (as 10 lies between these two numbers)

Step 2: Divide given number by one of those selected square roots.

Divide 10 by 3.

=> 10/3 = 3.33 (round off answer at 2 places)

Step 3: Find the average of root and the result from the above step i.e.

(3 + 3.33)/2 = 3.1667

Verify: 3.1667 x 3.1667 = 10.0279 (Not required)

Repeat step 2 and step 3

Now 10/3.1667 = 3.1579

Average of 3.1667 and 3.1579.

(3.1667+3.1579)/2 = 3.1623

Verify: 3.1623 x 3.1623 = 10.0001 (more accurate)

Example 2: Find the square roots of whole numbers perfect squares from 1 to 100.

Solution: The perfect squares from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

 Square root Result √1 1 √4 2 √9 3 √16 4 √25 5 √36 6 √49 7 √64 8 √81 9 √100 10

Example 3: What is,

1. Square root of 2
2. Square root of 3
3. Square root of 4
4. Square root of 5

Solution: Use square root list, we have

1. value of root 2 i.e. √2 = 1.4142
2. value of root 3 i.e. √3 = 1.7321
3. value of root 4 i.e. √4 = 2
4. value of root 5 i.e. √5 = 2.2361

Example 4: Is square Root of a Negative Number a whole number?

Solution: No, As per the square root definition, negative numbers shouldn’t have a square root. Because if we multiply two negative numbers result will always be a positive number.  Square roots of negative numbers expressed as multiples of i (imaginary numbers).

Practice Problems:

1. Simplify √142
2. Find the value of √12.
3. Are √155, √121 and √139 perfect squares?

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