 # Chapter 3: Squares and Square Roots Exercise – 3.3

### Question: 1

Find the squares of the following numbers using column method. Verify the result finding the square using the usual multiplication.

(i) 25

(ii) 37

(iii) 54

(iv) 71

(v) 96

### Solution:

Here a = 2, b = 5

Step: 1 Make 3 columns and write the values of a2, 2 x a x b, and b2 in these columns.

Sol.

(i) 25

Here a = 2, b = 5

Step: 1 Make 3 columns and write the values of a2, 2 x a x b, and b2 in these columns.

 Column I Column II Column III a2 2 x a x b b2 4 20 25

Step: 2 Underline the unit digit of b2 (in Column III) and add its tens digit, if any, with 2 x a x b (in column II)

 Column I Column II Column III a2 2 x a x b b2 4 20 + 2 25 22

Step: 3 Underline the unit digit in Column II and add the number formed by the tens and other digits if any, with a2 in Column I.

 Column I Column II Column III a2 2 x  a x b b 4 + 2 20 + 2 25 6 22

Step 4: Underline the number in Column I.

 Column I Column II Column III a2 2 x a x b b2 4 + 2 20 + 2 25 6 22

Step: 5 write the underlined digits at the bottom of each column to obtain the square of the given number.

In this case, we have:

252 = 625

Using Multiplication:

25 x 25 = 625

This matches with the result obtained by the column method:

(ii) 37

Here, a = 3, b = 7

Step: 1 Make 3 columns and write the values of a2, 2 x a x b, and b2 in these columns.

 Column I Column II Column III a2 2 x a x b b2 9 42 49

Step: 2 Underline the unit digit of b2 (in Column III) and add its tens digit, if any, with 2 x a x b (in Column II)

 Column I Column II Column III A2 2 x a x b B2 9 + 4 42 + 4 49 13 46

Step: 3 Underline the unit digit in Column II and add the number formed by tens and others digits if any, with a2 in Column I.

 Column I Column II Column III A2 2 x a x b B2 9 + 4 42 + 4 49 13 46

Step 4:  Write the underlined digits at the bottom of each column to obtain the square of the given number.

In this case, we have:

372 = 1369

Using multiplication:

37 x 37 = 1369

This matches with the result obtained using the column method.

(iii) 54

Here, a = 5, b = 4

Step 1: make 3 columns and write the values of a2, 2 x a x b and b2 in these columns.

 Column I Column II Column III a2 2 x a x b b2 25 40

Step: 2 Underline the unit digit of b2 (in Column III) and add its tens digit, if any, with 2x a x b (in Column II)

 Column I Column II Column III a2 2 x a x b b2 25 40 +1 16 41

Step: 3 Underline the digit in Column II and add the number formed by the tens and other digits if any, with a2 in Column I.

 Column I Column II Column III a2 2 x a x b b2 25 + 4 40 + 1 16 29 41

Step: 4 underline the number in Column I.

 Column I Column II Column III a2 2 x a x b b2 25 + 4 40 + 1 16 29 41

Step: 5 write the underlined digits at the bottom of each column to obtain the square of the given number.

In this case, we have:

542 = 2916

Using multiplication:

54 x 54 = 2916

This matches with the result obtained using the column method.

(iv) 71

Here, a = 7, b = 1

Step: 1 Make 3 columns and write the values of a2, 2 x a x b and b2 in these columns.

 Column I Column II Column III a2 2 x a x b b2 49 14 1

Step: 2 Underline the unit digit of b2 (in column III) and add its ten digit, if any with 2 x a x b (in column II)

 Column I Column II Column III a2 2 x a x b b2 49 14 + 0 1 14

Step: 3 Underline the unit digit in Column II and add the number formed by the tens and other digits, if any, with a2 in column I.

 Column I Column II Column III a2 2 x a x b b2 49 + 1 14 + 0 1 50 14

Step: 4 underline the number in column I.

 Column I Column II Column III a2 2 x a x b b2 49 + 1 14 + 0 1 50 14

Step: 5 write the underlined digits at the bottom of each column to obtain the square of the given number:

In this case, we have:

712 = 5041

Using multiplication:

71 x 71 = 5041

This matches with the result obtained using the column method.

(v) 96

Here, a = 9, b = 6

Step: 1 Make 3 columns and write the values of a2, 2 x a x b and b2 in these columns.

 Column I Column II Column III a2 2 x a x b b2 81 108 36

Step: 2 Underline the unit digit of b2 (in column III) and add its tens digit, if any with 2 x a x b (in column II)

 Column I Column II Column III a2 2 x a x b b2 81 108 + 3 36 111

Step: 3 Underline the unit digit in Column II and add the number formed by the tens and other digits if any, with a2 in column I.

 Column I Column II Column III a2 2 x a x b b2 81 + 11 108 + 3 36 92 111

Step: 4 underline the number in Column I

 Column I Column II Column III a2 2 x a x b b2 81 + 11 108 + 3 36 92 111

Step: 5 write the underlined digits at the bottom of each column to obtain the square of the given number.

In this case, we have:

962 = 9216

Using multiplication:

96 x 96 = 2916

This matches with the result obtained using the column method.

### Question: 2

Find the squares of the following numbers using diagonal method:

(i) 98

(ii) 273

(iii) 348

(iv) 295

(v) 171

### Solution:

(i) 98

∴ 982 = 9604 (ii) 273

∴ 2732 = 74529 (iii) 348

∴ 3482 = 121104 (iv) 295

∴ 2952 = 87025 (v) 171

∴ 1712 = 29241 ### Question: 3

Find the squares of the following numbers:

(i) 127

(ii) 503

(iii) 451

(iv) 862

(v) 265

### Solution:

We will use visual method as it is the efficient method to solve this problem.

(i) We have:

127 = 120 + 7

Hence, let us draw a square having side 127 units. Let us split it into 120 units and 7 units. Hence, the square of 127 is 16129.

(ii) We have:

503 = 500 + 3

Hence, let us draw a square having side 503 units. Let us split it into 500 units and 3 units. Hence, the square of 503 is 253009.

(iii) We have:

451 = 450 + 1

Hence, let us draw a square of having side 451 units. Let us split it into 450 units and 1 units. Hence, the square of 451 is 203401.

(iv) We have:

862 = 860 + 2

Hence, let us draw a square having side 862 units. Let us split it into 860 units and 2 units. Hence, the square of 862 is 743044.

(v) We have:

265 = 260 +5

Hence, let us draw a square having 265 units. Let us split it into 260 units and 5 units. Hence, the square of 265 units is 70225.

### Question: 4

Find the squares of the following numbers:

(i)  425

(ii) 575

(iii) 405

(iv) 205

(v) 95

(vi) 745

(vii) 512

(viii)  995

Notice that all numbers except the one in question (vii) has 5 as their respective unit digits. We know that the square of a number with the form n5 is a number ending with 25 and has the number n(n + 1) before 25.

### Solution:

Sol.

(i)  425

Here, n = 42

∴ n(n + 1) = (42)(43) = 1806

∴ 4252 = 180625

(ii) 575

Here, n = 57

∴n(n + 1) = (57)(58) = 3306

∴ 5752 = 330625

(iii) 405

Here n = 40

∴n(n + 1) = (40)(41) = 1640

∴ 4052 = 164025

(iv) 205

Here n = 20

∴n(n + 1) = (20)(21) = 420

∴ 2052 = 42025

(v) 95

Here n = 9

∴ n(n + 1) = (9)(10) = 90

∴ 952 = 9025

(vi) 745

Here n = 74

∴ n(n + 1) = (74)(75) = 5550

∴ 7452 = 555025

(vii) 512

We know: The square of a three-digit number of the form 5ab = (250 + ab) 1000 + (ab)2

∴ 5122 = (250+12)1000 + (12)2 = 262000 + 144 = 262144

(viii)  995

Here, n = 99

∴ n(n + 1) = (99)(100) = 9900

∴ 9952 = 990025

### Question: 5

Find the squares of the following numbers using the identity (a +b)2 = a2 + 2ab + b2:

(i) 405

(ii) 510

(iii) 1001

(iv) 209

(v) 605

### Solution:

(i) 405

On decomposing:

405 = 400 + 5

Here, a = 400x and b = 5

Using the identity (a + b)2 = a2 + 2ab + b2:

4052 = (400 + 5)2= 4002 + 2(400)(5) + 52 = 160000 + 4000 + 25 = 164025

(ii) 510

On decomposing:

510 = 500 + 10 Here, a = 500 and b = 10

Using the identity (a + b)2 = a2 + 2ab + b2:

5102 = (500 + 10)2 = 5002 + 2(500)(10) + 102 = 250000 + 10000 + 100 = 260100

(iii) 1001

On decomposing:

1001 = 1000 + 1

Here, a = 1000 and b = 1

Using the identity (a + b)2 = a2 + 2ab + b2:

10012 = (1000 +1)2 = 10002 + 2(1000) (1) + 12 = 1000000 + 2000 + 1 = 1002001

(iv) 209

On decomposing:

209 = 200 + 9

Here, a = 200 and b = 9

Using the identity (a + b) 2 = a2+ 2ab + b2:

2092 = (200 + 9)2 = 2002 + 2(200)(9) + 92 = 40000 + 3600 + 81 = 43681

(v) 605

On decomposing:

605 = 600 + 5

Here, a = 600 and b = 5

Using the identity (a + b)2 = a2 + 2ab + b2:

6052 = (600 + 5)2 = 6002 + 2(600)(5) + 52 = 360000 + 6000 + 25 = 366025

### Question: 6

Find the squares of the following numbers using the identity (a – b)2 = a2  – 2ab + b2:

(i) 395

(ii) 995

(iii) 495

(iv) 498

(v) 99

(vi) 999

(vii) 599

### Solution:

(i) 395

Decomposing: 395 = 400 - 5

Here, a = 400 and b = 5

Using the identity (a — b) 2 = a2 – 2ab + b2:

3952 = (400 - 5)2 = 4002 - 2(400)(5) + 52 = 160000 - 4000 + 25 = 156025

(ii) 995

Decomposing:

995 = 1000 - 5

Here, a =1000 and b = 5

Using the identity (a -b) 2 = a2 - 2ab + b2:

9952 = (1000 - 5)2 = 10002 -2(1000)(5) + 52 = 1000000 -10000 + 25 = 990025

(iii) 495

Decomposing: 495 = 500 - 5 Here, a = 500 and b = 5 Using the identity (a - b) 2 = a2 - 2ab + b2:

4952 = (500 - 5)2 = 5002 - 2(500) (5) + 52 = 250000 - 5000 + 25 = 245025

(iv) 498

Decomposing: 498 = 500 - 2 250000 - 5000 + 25 = 245025

Here, a = 500 and b = 2

Using the identity (a - b)2 = a2 - 2ab + b2:

4982 = (500 -2)2 = 5002 -2(500)(2) + 22 = 250000 - 2000 + 4 = 248004

(v) 99

Decomposing: 99 = 100 -1 Here, a = 100 and b = 1 Using the identity (a - b)2 = a2 - 2ab + b2: 992 = (100-1)2 = 1002- 2(100)(1) + 12 = 10000 -200 + 1 = 9801

(vi) 999

Decomposing: 999 = 1000 – 1

Here, a = 1000 and b = 1

Using the identity (a - b)2 = a2 - 2ab + b2:

9992 = (1000 – 1)2 – 10002 – 2(1000) (1) + 12 = 1000000 – 2000 +1 = 998001

(vii) 599

Decomposing: 599 = 600 -1

Here, a = 600 and b = 1

Using the identity (a - b) 2 = a2 -2ab + b2:

5992 = (600 -1)2 = 6002-2(600) (1) + 12 = 360000 -1200 + 1 = 358801

### Question: 7

Find the squares of the following numbers by visual method:

(i) 52

(ii) 95

(iii) 505

(iv) 702

(v) 99

### Solution:

(i) 52

We have:

52 = 50 + 2

Let us draw a square having side 52 units. Let us split it into 50 units and 2 units. The sum of the areas of these four parts is the square of 52. Thus, the square of 52 is 2704.

(ii) 95

We have:

95 = 90 + 5

Let us draw a square having side 95 units. Let us split it into 90 units and 5 units. The sum of the areas of these four parts is the square of 95. Thus, the square of 95 is 9025.

(iii) 505

We have:

505 = 500 + 5

Let us draw a square having side 505 units. Let us split it into 500 units and 5 units. The sum of the areas of these four parts is the square of 505. Thus, the square of 505 is 255025.

(iv) 702

We have:

702 = 700 + 2

Let us draw a square of having side 702 units. Let us split it into 700 units and 2 units. The sum of the areas of these four parts is the square of 702. Thus, the square of 702 is 492804.

(v) 99

We have:

99 = 90 + 9

Let us draw a square of having side 99 units. Let us split it into 90 units and 9 units. The sum of the areas of these four parts is the square of 99. Thus, the square of 99 is 9801. ### Course Features

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