Finding the Square of a Number

Finding the Square of a Number is a simple method. We need to multiply the given number by itself to find its square number. The square term is always represented by a number raised to the power of 2. For example, the square of 6 is 6 multiplied by 6, i.e., 6×6 = 62 = 36.

Thus, to find the square of single-digit numbers, we can simply multiply them by itself. Also, by remembering the tables from 1 to 10, we can quickly find the square of the number.

Number

Square of Number

1

1

2

4

3

9

4

16

5

25

6

36

7

49

8

64

9

81

10

100

But what if we have to find the square of a two-digit number? It could be a little tricky. We cannot find the square of two-digit numbers easily, by simple multiplication. In this article we can learn how to find the square of such numbers.

How to Calculate the square of a number?

Here we are going to find the square of a natural number by expanding the square. Let us learn with examples.

Example: Find the square of 24.

Solution: We can write 24 as:

24 = (20 + 4)

Put square on both the sides:

242 = (20 +4)2

= (20 + 4) (20 + 4)

Open the brackets and multiply the terms.

= 20 x 20 + 20 x 4 + 4 x 20 + 4 x 4

Now we can simplify the above expression easily.

= 400 + 80 + 80 + 16

= 576

Note: We can observe that the expression of (20 + 4)2 is similar to the algebraic expression (a+b)2 which is equal to:

(a+b)2 = a2 + 2ab +b2

Therefore, we can use the same formula to find the square root of numbers.

Finding the Square Using Patterns

While squaring the numbers, we may find certain patterns that will help us remember the squares. Let us see some patterns:

  • 252 = 625 = 600 + 25 = 6 x 100 + 25 = (2 × 3) hundreds + 25
  • 352 = 1225 = 1200 + 25 = 12 x 100 + 25 = (3 × 4) hundreds + 25
  • 752 = 5625 = 5600 + 25 = 56 x 100 + 25 = (7 × 8) hundreds + 25
  • 1252 = 15625 = 15600 + 25 = 156 x 100 + 25 = (12 × 13) hundreds + 25

From the above pattern, we can see all the numbers that are squared have 5 at their unit’s place. Now, say n5 is a number that is squared. Thus, by looking at the above patterns, we can write the generalised expression.

(n5)2 = (10n + 5)2

= 10n(10n + 5) + 5(10n + 5)

= 100n2 + 50n + 50n + 25

= 100n(n + 1) + 25

= n(n + 1) hundred + 25

Hence, the shortcut to find the square of numbers having 5 at their unit’s place is:

(n5)2 = n(n + 1) hundred + 25

Example: Find the square of 95.

Solution: Given the number is 95.

Square of 95 = 952

Since, n = 9

Therefore using the above pattern to find the square of a number with 5 at unit place, we get;

(n5)2 = n(n + 1) hundred + 25

Here n = 9

(95)2 = 9 (9+1) hundreds + 25

= 9×10 hundreds + 25

= 9000 + 25

= 9025

Thus, the square of 95 is 9025.

Squares and Square Root Related Articles

Finding Square of Number Using Pythagorean triplets form

While learning about the right triangles, we have understood that, by Pythagoras theorem, we can find the length of any side of the triangle, given the other two sides.

The three sides of the right triangle are hypotenuse, perpendicular and base. As per Pythagoras theorem,

Hypotenuse2 = Perpendicular2 + Base2

Suppose, the length of the sides are given as:

Perpendicular = 3

Base = 4

Hypotenuse = 5

Now, if we analyse, the square of the hypotenuse is equal to the sum of squares of perpendicular and base.

52 = 32 + 42

25 = 9 + 16

25 = 25

Thus, we can conclude that 3, 4 and 5 are Pythagorean triplets.

Another example of Pythagorean triplets are 6, 8 and 10.

102 = 62 + 82

100 = 36+ 64

100 = 100

Thus, by the above two examples, we can generalise the form of Pythagorean triplet.

Suppose n is any natural number, such that n > 1, we have;

(2n)2 + (n2 – 1)2 = (n2 + 1)2

Or we can say, for any natural number, we know that 2n, n2 – 1 and n2 + 1 are Pythagorean triplets.

Solved Examples

Q.1: Find the square of the following numbers:

(i) 86

(ii) 71

(iii) 55

(iv) 95

Solution:

(i) 86

We can write the given number as:

86 = (80 + 6)

Squaring both the sides, we get;

862 = (80 + 6)2

= (80 + 6) (80 + 6)

Expanding the brackets.

= 80 x 80 + 80 x 6 + 6 x 80 + 6×6

= 6400 + 480 + 480 + 36

= 7396

(ii) 71

We can write the given number as:

71 = 70 + 1

Squaring both the sides, we get;

712 = (70+1)2

= (70 + 1) (70 + 1)

Expanding the brackets.

= 70 x 70 + 70 x 1 + 1 x 70 + 1 x 1

= 4900 + 70 + 70 + 1

= 5041

(iii) 55

We can write the given number as:

55 = 50 + 5

Squaring both the sides, we get;

552 = (50 + 5)2

= (50 + 5) (50 +5)

Expanding the brackets.

= 50 x 50 + 50 x 5 + 5 x 50 + 5 x 5

= 2500 + 250 + 250 + 25

= 2500 +500 +25

= 3025

(iv) 95

We can write the given number as:

95 = 90 + 5

Squaring on both the sides, we get;

952 = (90+5)2

= (90 +5) (90+5)

Expanding the brackets.

= 90 x 90 + 90 x 5 + 5 x 90 + 5 x 5

= 8100 + 450 + 450 + 25

= 9025

Q.2: Write a Pythagorean triplet whose one member is:

(i) 6

(ii) 18

Solution: For any natural number, we know that 2n, n2 – 1 and n2 + 1 are Pythagorean triplets.

(i) 6

Let, 2m = 6

⇒ m = 6/2 = 3

m2–1= 32 – 1 = 9–1 = 8

m2+1= 32+1 = 9+1 = 10

Therefore, (6, 8, 10) is a Pythagorean triplet.

(ii) 18

Let,

2m = 18

⇒ m = 18/2 = 9

m2–1 = 92–1 = 81–1 = 80

m2+1 = 92+1 = 81+1 = 82

Therefore, (18, 80, 82) is a Pythagorean triplet.

Q.3: Find if (8, 15, 17) is a Pythagorean triplet.

Solution: Given, (8, 15, 17)

LHS = 82 + 152 = 289

RHS = 172 = 289

LHS = RHS

Hence, the given triplet is Pythagorean.

Q.4: Find the square of 205 using the pattern of squares method.

Solution: By the pattern of squares, we know that, if a number has 4 at its unit’s place, then the square of the number can be determined by:

(n5)2 = n(n + 1) hundred + 25

Here, n = 20

Therefore,

(205)2 = 20 (20+1) hundreds + 25

= 20 x 21 hundreds + 25

= 420 x 100 + 25

= 42000+25

= 42025

Therefore, the square of 205 is 42025

Practice Questions

  1. Find the square of:
    1. 44
    2. 52
    3. 89
    4. 76
  2. Write a Pythagorean triplet whose smallest member is 8.
  3. Find a Pythagorean triplet in which one member is 10.
  4. Find the squares of the following numbers containing 5 in the unit’s place.
    1. 15
    2. 105

Frequently Asked Questions on Finding the square

How to find the square of a single-digit number?

To find the square of a single-digit number, multiply the original number by itself. For example, a square of 4 = 42 = 4 x 4 = 16.

How to find the square of a two-digit number?

We can split the given number into two parts such that one part is the multiples of 10 and then square the sum of two parts. Separate and expand the brackets. Simplify the expression to get the square of the original number. For example, the square of 38 is:
382 = (30 + 8)2
= (30+8) (30 +8)
= 30 x 30 + 30 x 8 + 8 x 30 + 8 x 8
= 900 + 240 + 240 + 64
= 1444

How to find the square of a number with 5 at the unit place?

If the number has 5 at its unit place, then to find the square of such numbers, we can use (n5)2 = n(n + 1) hundred + 25, where n is any natural number. To find the square of 35, put n = 3.

What is the smallest number of three digits which is a perfect square?

The smallest number of three digits which is a perfect square is 100 because 102 = 10 x 10 = 100.

What will be the unit’s digit of the square of 98?

The square of unit digit of 98 is 8 x 8 = 16. Therefore, the unit digit of square of 98 will be 6.

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