**Squares and Square Roots** are commonly used in solving mathematical problems. Remembering and applying these concepts of **Squares and Square Roots** quickly in problem-solving techniques give trade-offs in competitive exams.

When a positive number is multiplied by itself then the result is said to be the **Square** of that number, on the other hand, if the **Square Root** of the result and it will result in the same positive number. For Example, if x is the square root of y, then it’s shown as x = √y or it can be shown as x^{2} = y.

**Numeric Example**: If 16 is the square root of 4, then it can be shown as 4 = √16 or it can be shown as 4^{2} = 16. It’s easier to find out squares of the given number but it requires effort and the right method to find out the Square Root of any given number which is not the square of a perfect number.

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## Squares

The Square of a number is the product of the multiplication of the number twice. For Example: If a number is expressed as ‘x’ then the square of ‘x’ is shown as x multiplied by x and the product is **x ^{2}**. Square Numbers are also called

**Perfect Numbers**.

The product of numbers which is not the same is not called a perfect square number. For example: 5 x 6 = 30, 30 is not a perfect number. Whereas, 5 x 5 = 25, hence 25 is called a **perfect square** number.

### First 50 Number Squares

1^{2} = 1 | 11^{2} = 121 | 21^{2} = 441 | 31^{2} = 961 | 41^{2} = 1681 |

2^{2} = 4 | 12^{2} = 144 | 22^{2} = 484 | 32^{2} = 1024 | 42^{2} = 1764 |

3^{2} = 9 | 13^{2} = 169 | 23^{2} = 529 | 33^{2} = 1089 | 43^{2} = 1849 |

4^{2} = 16 | 14^{2} = 196 | 24^{2} = 576 | 34^{2} = 1156 | 44^{2} = 1936 |

5^{2} = 25 | 15^{2} = 225 | 25^{2} = 625 | 35^{2} = 1225 | 45^{2} = 2025 |

6^{2} = 36 | 16^{2} = 256 | 26^{2} = 676 | 36^{2} = 1296 | 46^{2} = 2116 |

7^{2} = 49 | 17^{2} = 289 | 27^{2} = 729 | 37^{2} = 1369 | 47^{2} = 2209 |

8^{2} = 64 | 18^{2} = 324 | 28^{2} = 784 | 38^{2} = 1444 | 48^{2} = 2304 |

9^{2} = 81 | 19^{2} = 361 | 29^{2} = 841 | 39^{2} = 1521 | 49^{2} = 2401 |

10^{2} = 100 | 20^{2} = 400 | 30^{2} = 900 | 40^{2} = 1600 | 50^{2} = 2500 |

### Shortcut to find Squares

It’s tricky to find squares of numbers greater than 50, this process is a bit time-consuming and also it’s hard to memorize all Square Numbers greater than 50. To find short ways for squares of big numbers, we are dividing these numbers into 3 categories and a trick for each category.

#### Square of Numbers greater than 50

Let’s understand a **shortcut ways on Squares and Square Roots to find the Squares of numbers greater than 50** with the help of the below example:**Example**: **Find Square of 53?**

=> (53)^{2} => (50/2) + 3 and (3)^{2}

=> 25 + 3 and 09 {**Note**: for single digit we’ll add 0 before square}

=> 2809

#### Square of Numbers greater than 100

Let’s understand a **shortcut ways on Squares and Square Roots to find Squares of numbers greater than 100** with the help of the below example:**Example**: **Find a Square of 104?**

=> (104)^{2} => (104 + 4) and (4)^{2}

=> 108 and 16

=> 10816

**Example**: **Find Square of 112?**

=> (112)^{2} => (112 + 12) and (12)^{2}

=> 124 and 144

=> 124 + 1 and 44 {**Note**: For number greater than 99 we’ll add it with sum.}

=> 12544

#### Square of Numbers greater than 150

Let’s understand a **shortcut way on Squares and Square Roots to find Squares of numbers greater than 150** with the help of the below example:

Example: Find a **Square of 156?**

=> (156)^{2} => 3 x {(150/2) + 6} and (6)^{2}

=> 3 x {75 + 6} and 36

=> 3 x {81} and 36

=> 243 and 36

=> 24336

**Example**: **Find Square of 146?**

=> (146)^{2} => 3 x {(150/2) – (150 – 146)} and (150 – 146)^{2}

=> 3 x {(75) – (4)} and (4)^{2}

=> 3 x {71} and (16)^{2}

=> 213 and 16

=> 21316

### Characteristics of Square Numbers

- No Square Numbers will have unit place digits as 2, 3, 7, or 8.
- If Square Number is having an even number of zeros then the square root of that number will be the perfect number.
- A Square of an even number will always be an even number, whereas the odd number square will be odd.

## Square Roots

**Square Roots** symbol is ‘* √*‘ and it’s called a radical symbol. If we’re trying to find the square root of a number x then this number is called the radicand. It’s expressed as

*x.*

**√**### How to find Square Roots?

To find the square root of any given number we must check first if the given number is a perfect square or not. If the given number is a **perfect square** number then we can easily find that number. If the number is not a perfect square number then we can use the **Long Division Method** to find out the square root of that number.

#### Square root of a number by Long Division Method

Let’s understand the Long Division Method for finding out the square root of 1,04,976. Below are the steps to **find the square root of 104976 by the long division method**:

**Step 1**: Separate the digits by placing commas from right to left once in two digits pairs. for example: 10, 49, 76.**Step 2**: Now, find a number square that is less than or equal to the first two digits of the number (i.e. 10). For number 10 we find the perfect square of 3. A Square of 3 is 9 which is less than 9. So in this division: Remainder is 1, Divisor and Quotient are 3.**Step 3**: Now we’ll bring down 49, and the new number will be 149 which new Dividend.**Step 4**: Now we’ll find Divisor for 149, we’ll get the first digit from the last quotient multiplied by 2 (i.e. 6).**Step 5**: Now, we’ll find a unit place of the quotient in such a way that 6a multiplied by a will give product less than or equal to 149. For example, if a is 2 then 62 x 2 = 124.**Step 6**: Upon Dividing 149 by 62 we get the remainder as 25.**Step** 7: Now we’ll bring down 76, and the new number will be 2576 which new Dividend.**Step 8**: Now we’ll find Divisor for 2576, we’ll get the first digits from the last quotient multiplied by 2 (i.e. 32 x 2 = 64).**Step 9**: Now, we’ll find a unit place of the quotient in such a way that 64a multiplied by a will give product less than or equal to 2576. For example, if a is 4 then 64 x 4 = 2576.

Hence, the square root that we found by long division method is 324.

### First 50 Numbers Square Root

√1 = 1 | √11 = 3.3166 | √21 = 4.5826 | √31 = 5.5678 | √41 = 6.4031 |

√2 = 1.4142 | √12 = 3.4641 | √22 = 4.6904 | √32 = 5.6569 | √42 = 6.4807 |

√3 = 1.7321 | √13 = 3.6056 | √23 = 4.7958 | √33 = 5.7446 | √43 = 6.5574 |

√4 = 2 | √14 = 3.7417 | √24 = 4.899 | √34 = 5.831 | √44 = 6.6332 |

√5 = 2.2361 | √15 = 3.873 | √25 = 5 | √35 = 5.9161 | √45 = 6.7082 |

√6 = 2.4495 | √16 = 4 | √26 = 5.099 | √36 = 6 | √46 = 6.7823 |

√7 = 2.6458 | √17 = 4.1231 | √27 = 5.1962 | √37 = 6.0828 | √47 = 6.8557 |

√8 = 2.8284 | √18 = 4.2426 | √28 = 5.2915 | √38 = 6.1644 | √48 = 6.9282 |

√9 = 3 | √19 = 4.3589 | √29 = 5.3852 | √39 = 6.245 | √49 = 7 |

√10 = 3.1623 | √20 = 4.4721 | √30 = 5.4772 | √40 = 6.3246 | √50 = 7.0711 |

## Squares and Square Roots Use

**Application of Squares and Square Roots** is in solving many Quadratic equations, Trigonometric and mensuration equations. However, **Squares and Square Roots** are the basic building blocks of mathematics, and having expertise in numerical calculations will always be helpful in solving complex questions.

Memorizing Squares and Square Roots to 50 numbers are very essential so that we don’t waste time in solving Squares and Square Roots.

Learn Surds and Indices which have the direct application of **Squares and Square Roots**.

## FAQs

### What is a square root?

A square root is a product of a number when multiplied by itself.

### How to find square root?

Square root can be found in two methods: **Prime factorization** and the **Long division method**.

### Which method is best for finding the square root of a big number?

The **long division method** is perfect for finding the square root of big numbers.

### Application of Squares and Square Roots?

Squares and Square Roots can be used in Geometry, Quadratic equations, Calculus, etc.

### Which method is best for finding the square root of a small number?

Prime factorization is perfect for finding the square root of small numbers.