Square of any Number

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Maths Tricks and Shortcuts to Find the Square of any Number

Squaring of a number is used in mathematical calculations. In competitive exams, calculation speed is very crucial  to complete the exam on time. You can find many rules to find the square of a number. So, here you will find maths tricks and shortcuts to find the square of any number.

Case 1: Square of a number ending with 5

Let the given number be A5.

Step 1: In finding the last two digits of the answer, find the square of 5.

Step 2: Find  A + 1.

Step 3: Multiply  A and (A + 1). The result so obtained will be the left –most digit of the answer.

Let us now take some examples for your better understanding of the Maths Tricks and Shortcuts to find the square of any number ending with 5.

Example 1: Find the square of 25.

Maths Trick and Shortcut:

Square of 25

= Product of 2 and 3 I Square of 5

= 6 I 25

= 625

Example 2: Find the square of 75.

Maths Trick and Shortcut:

Square of 75

= Product of 7 and 8 I Square of 5

= 56 I 25

= 5625

Example 3: Find the square of 95.

Maths Trick and Shortcut:

Square of 95

= Product of 9 and 10 I Square of 5

= 90 I 25

= 9025

Note:  The same method can be extended to three-digit numbers.

Example 1: Find the square of 155.

Maths Trick and Shortcut:

Square of 155

= Product of 15 and 16 I Square of 5

= 240 I 25

= 24025

Example 2: Find the square of 165.

Maths Trick and Shortcut:

Square of 165

= Product of 16 and 17 I Square of 5

= 272 I 25

= 27225

Example 3: Find the square of 105.

Maths Trick and Shortcut:

Square of 105 = Product of 10 and 11 I Square of 5

= 110 I 25

= 11025

Now, try the following questions:

  • 85 × 85
  • 65 × 65
  • 195 × 195
  • 125 × 125


Case 2: Square of any two- digit number

Step 1: In finding the last two digits of the answer, find the square of the last digit of the given number.

Step 2: Multiply the two digits of the given number together and double it.

Step 3: In finding the first two digits of the answer, find the square of  left hand digit of the given number.

Let us now take some examples for your better understanding of the Maths Tricks and Shortcuts to find the square of any two-digit number.

Example 1: Find the square of 26.

Maths Trick and Shortcut:

Square of 26

= Square of 2 I Double the product of 2 and 6 I Square of 6

= 4 I 2 × 2 × 6 I 36

= 4 I 24 I 36

Collapsing the numbers

= 4 I 24 + 3 I 6

= 4 I 27 I 6

=  4 + 2  I 7 I 6

= 6  I 7 I 6

Hence, the answer is 676. 

Example 2: Find the square of 48.

Maths Trick and Shortcut:

Square of 48

= Square of 4 I Double the product of 4 and 8 I Square of 8

= 16 I 2 × 4 × 8 I 64

= 16 I 64 I 64

Collapsing the numbers

= 16 I 64 + 6 I 4

= 16 I 70 I 4

=  16 + 7  I 0 I 4

= 23  I 0 I 4

Hence, the answer is 2304.

Example 3: Find the square of 97.

Maths Trick and Shortcut:

Square of 97

= Square of 9 I Double the product of 9 and 7 I Square of 7

= 81 I  2 × 9 × 7 I 49

= 81 I 126 I 49

Collapsing the numbers

= 81 I 126 + 4 I 9

= 81 I 130 I 9

=  81 + 13  I  0  I 9

= 94  I 0 I 9

Hence, the answer is 9409.

Now, try the following questions:

  • 89 × 89
  • 63 × 63
  • 19 × 19

 

Case 3: Square of a number which is nearer to 10n

 In order to find the square of a number which is nearer to 10n, we use the following algebraic formula:

                       x2 = (x2 – y2) + y2 = ( x + y)( x -  y) + y2

Let us now take some examples for your better understanding of the Maths Tricks and Shortcuts to find the square of any number, which is nearer to 10n.

Example 1: Find the square of 98.

Maths Trick and Shortcut:

Square of 98 = (98 + 2)( 98 – 2 ) + 22 = 9600 + 4 = 9604

Example 2: Find the square of 102.

Maths Trick and Shortcut:

Square of 102 = (102 + 2)( 102 – 2 ) + 22 = 10400 + 4 = 10404

Example 3: Find the square of 994.

Maths Trick and Shortcut:

Square of 994 = (994 + 6)( 994 – 6 ) + 62 = 988000 + 36 =988036

Now, try the following questions:

  • 97 × 97
  • 993 × 993
  • 1008 ×1008

 Note:

  • If there are n digits in a number, the square will have either 2n or 2n- 1 digits.
  • We can also check if our calculation in squaring is correct by using the digit-sum method.

Digit-sum method

   Step 1: Find the digit-sum of  RHS and LHS.

   Step 2: If the digit-sum of RHS is equal to the digit-sum of LHS, then our calculation is correct.

 Example 1: (897)2 = 804609

  Digit-Sum: (6)2 = 27

⇒ 36 = 27

⇒ 9 = 9

 Thus, our calculation is correct. 

Example 2: (98)2 = 9604

Digit-Sum: (8)2 = 19

⇒ 64 = 19

⇒ 10 = 10

⇒ 1 = 1

 Thus, our calculation is correct.

 Example 3: (207)2 = 42849

 Digit-Sum: (9)2 = 27

⇒ 81 = 9

⇒ 9 = 9

 Thus, our calculation is correct.

 Please comment below if you find any problem in any of the above given maths tricks and shortcuts to find the square of any number.