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. The Square root meaning of a number can be simply be defined as a number that has an equivalent value to two numbers multiplied by themselves. The numbers are usually the same when multiplied with one another and the square root is the number taken from it.

**Table of Contents:**

- Square Root Definition
- Square root Symbol
- Square roots Chart
- How do you figure out square roots?
- Examples
- Applications
- Practice Problems

## Square Root Definition

Square root of any number gives same number when multiplied by itself.

For example, √(m X m) = √(m^{2}) = 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*

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 y^{2} = a, where multiplying the number y with itself will lead to its square root (y.y).

#### Square Root Formula

Using the above explanation, we can conclude that,

y.y = y^{2} = a ; where ‘a’ is the square root of a number ‘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:

*‘ √x ‘ *where x is the number itself.

## Square Roots Chart

Square root chart 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 |
4.8990 | √41 |
6.4031 |

√8 |
2.8284 | √25 |
5 | √42 |
6.4807 |

√9 |
3 | √26 |
5.0990 | √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 |
3.8730 | √32 |
5.6569 | √49 |
7 |

√16 |
4 | √33 |
5.7446 | √50 |
7.0711 |

√17 |
4.1231 | √34 |
5.8310 |

### 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 = 7^{2} = 49

Square root of 49, **√**49 = 7

### How to find square root 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 procedure until you have the desired accurate result. Mostly used to find the square roots of numbers that aren’t perfect squares.

Let us understand this concept with the help of example:

Solve **√**10 to 2 decimal places.

Step 1: 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 **√**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!

### Examples

**Example 1**: 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 2**: 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

- Square root of 2 = 1.4142
- Square root of 3 = 1.7321
- Square root of 4 = 2
- Square root of 5 = 2.2361

**Example 3**: 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).

### Applications

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. 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.

### Practice Problems

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

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