The positive number, when multiplied by itself, represents the square of the number. The square root of square of a positive number gives the original number. Suppose, x is the square root of y, then it is represented as x=âˆšy or we can express the same equation as x^{2}Â = y. Here, ‘âˆš’ is the radical symbol used to represent the root of numbers.

For example, the square of 3 is 9, 3^{2} = 9 and the square root of 9,Â âˆš9 = 3. Since 9 is a perfect square, hence it is easy to find the square root of such numbers, but for an imperfect square like 3, 7, 5, etc., it is really tricky to find the root. Learn square roots from 1 to 25Â with some shortcut tricks here.

**Table of Contents:**

- Definition
- Symbol
- Formula
- Property
- Perfect Squares
- Table
- Finding Square Root
- Square Root of Complex Numbers
- Square Root Equation
- Squares and Square Root
- Applications
- Examples
- FAQs

## Definition

The square root of any number is equal to a number, which when squared gives the original number.

Let us say m is a positive integer, such thatÂ âˆš**(m.m) = âˆš(m ^{2}) = m**

**Note:** The square root of a negative number represents a complex number**.**

SupposeÂ âˆš-n = iâˆšn, where i is the imaginary number.

## Square Root Symbol

The square root symbol is usually denoted as â€˜* âˆšâ€™*. It isÂ called a radical symbol. To represent a number â€˜xâ€™ as a square root using this symbol can be written as:

*â€˜*where x is the number itself. The number under the

**âˆšx**â€˜**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

The formula to find the square root is:

y = âˆša |

Since, y.y = y^{2}Â = a; where â€˜aâ€™ is the square of a number â€˜yâ€™.

## Property of Square root

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) = âˆšx**

Consider a number â€˜aâ€™ acting as the square root of a number â€˜yâ€™ such that y^{2} = a, where multiplying the number y with itself will lead to its square root (y.y).

### Perfect squares

Below are the numbers which are perfect squares and the finding the square roots of such numbers is easy.

- 1
^{2}= 1 - 2
^{2}= 4 - 3
^{2}= 9 - 4
^{2}= 16 - 5
^{2}= 25 - 6
^{2}= 36 - 7
^{2}= 49 - 8
^{2}= 64 - 9
^{2}= 81 - 10
^{2}= 100

Hence, 1,4,9,16,25,36,49,64,81 and 100 are the perfect squares here. Check square roots of some numbers here:

## Square Root Table

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

âˆšn |
Value | âˆšn |
Value | âˆšn |
Value |

âˆš1 |
1 | âˆš18 |
4.2426 | âˆš35 |
5.9161 |

âˆš2 |
1.4142 | âˆš19 |
4.3589 | âˆš36 |
6 |

âˆš3 |
1.7321 | âˆš20 |
4.4721 | âˆš37 |
6.0828 |

âˆš4 |
2 | âˆš21 |
4.5826 | âˆš38 |
6.1644 |

âˆš5 |
2.2361 | âˆš22 |
4.6904 | âˆš39 |
6.2450 |

âˆš6 |
2.4495 | âˆš23 |
4.7958 | âˆš40 |
6.3246 |

âˆš7 |
2.6458 | âˆš24 |
4.8990 | âˆš41 |
6.4031 |

âˆš8 |
2.8284 | âˆš25 |
5 | âˆš42 |
6.4807 |

âˆš9 |
3 | âˆš26 |
5.0990 | âˆš43 |
6.5574 |

âˆš10 |
3.1623 | âˆš27 |
5.1962 | âˆš44 |
6.6332 |

âˆš11 |
3.3166 | âˆš28 |
5.2915 | âˆš45 |
6.7082 |

âˆš12 |
3.4641 | âˆš29 |
5.3852 | âˆš46 |
6.7823 |

âˆš13 |
3.6056 | âˆš30 |
5.4772 | âˆš47 |
6.8557 |

âˆš14 |
3.7417 | âˆš31 |
5.5678 | âˆš48 |
6.9282 |

âˆš15 |
3.8730 | âˆš32 |
5.6569 | âˆš49 |
7 |

âˆš16 |
4 | âˆš33 |
5.7446 | âˆš50 |
7.0711 |

âˆš17 |
4.1231 | âˆš34 |
5.8310 |

## How do Find Square Root

To find the square root of any number, we need to figure out whether the given number is a perfect square or imperfect square. If the number is a perfect square, such as 4, 9, 16, etc., then we can factorise the number by prime factorisation method. If the number is imperfect square, such as 2, 3, 5, etc., then we have to use long division method to find the root.

Example: Square of 7 = 7 x 7 = 7^{2} = 49

The square root of 49, **âˆš**49 = 7

### 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 â€œguess 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.

## Square root of Complex Numbers

To find the square root of complex numbers is a little complicated process. We can find the square root of a+ib using the below formula:

**\(\sqrt{a+b i}=\pm(\sqrt{\frac{\sqrt{a^{2}+b^{2}+a}}{2}}+i \sqrt{\frac{\sqrt{a^{2}+b^{2}-a}}{2}})\)**

where a+ib is a complex number.

## 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 âˆš(4a+9) – 5 = 0**

Solution: Given, âˆš(4a+9) – 5 = 0 Isolate the square root term first. âˆš(4a+9) = 5 Now on squaring both the sides, we get; 4a+9 = 5^{2}

4a + 9 = 25 4a = 16 a = 16/4 a = 4

## Squares and Square Root

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

Numbers |
SquaresÂ |
Square root |

0 | 0^{2} = 0 |
âˆš0 = 0 |

1 | 1^{2} = 1 |
âˆš1 = 1 |

2 | 2^{2} = 4 |
âˆš4 = 2 |

3 | 3^{2} = 9 |
âˆš9 = 3 |

4 | 4^{2} = 16 |
âˆš16 = 4 |

5 | 5^{2} = 25 |
âˆš25 = 5 |

6 | 6^{2} = 36 |
âˆš36 = 6 |

7 | 7^{2} = 49 |
âˆš49 = 7 |

8 | 8^{2} = 64 |
âˆš64 = 8 |

9 | 9^{2} = 81 |
âˆš81 = 9 |

10 | 10^{2} = 100 |
âˆš100 = 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. Some of the applications are:

- Quadratic equations
- Algebra
- Geometry
- Calculus

## Solved Examples

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

**Example 1:Â Solve âˆš10 to 2 decimal places.**

**Solution:**

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

2^{2} = 4; 3^{2} = 9, 4^{2} = 16 and 5^{2} = 25

Choose 3 and 4 (as **âˆš**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)

Stop the process. Answer!

**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 |

âˆš1 |
1 |

âˆš4 |
2 |

âˆš9 |
3 |

âˆš16 |
4 |

âˆš25 |
5 |

âˆš36 |
6 |

âˆš49 |
7 |

âˆš64 |
8 |

âˆš81 |
9 |

âˆš100 |
10 |

**Example 3**: **What is:**

**Square root of 2****Square root of 3****Square root of 4****Square root of 5**

**Solution**: Use square root list, we have

- value of root 2 i.e. âˆš2 = 1.4142
- value of root 3 i.e. âˆš3 = 1.7321
- value of root 4 i.e. âˆš4 = 2
- value of root 5 i.e. âˆš5Â = 2.2361

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

**Solution**: No, As per the square root definition, negative numbers shouldnâ€™t have a square root. Because if we multiply two negative numbers result will always be a positive number. Â Square roots of negative numbers expressed as multiples of i (imaginary numbers).

### Practice Problems

- Simplify âˆš142
- Find the value of âˆš12.
- Are âˆš155, âˆš121 and âˆš139 perfect squares?

## Frequently Asked Questions â€“ FAQs

### What is a square root?

### How to find square root?

### What is squares and square roots?

### How to find square root of perfect squares?

### How to find the square root of imperfect squares?

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