Squares And Square Roots

(i) 441

In order to find if the given number is a perfect square,

At first,

We’ll resolve the given number into prime factors:

Hence,

441 = 49 × 9

= 7 × 7 × 3 × 3

= (7 × 3) × (7 × 3)

= 21 × 21

= (21)²

Hence, it is a perfect square.

(ii) 576

In order to find if the given number is a perfect square,

At first,

We’ll resolve the given number into prime factors:

Hence,

576 = 64 × 9

= 8 × 8 × 3 × 3

= 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3

= (2 × 2 × 2 × 3) × (2 × 2 × 2 × 3)

= 24 × 24

= (24)²

Hence, it is a perfect square.

(iii) 11025

In order to find if the given number is a perfect square,

At first,

We’ll resolve the given number into prime factors:

Hence,

11025 = 441 × 25

= 49 × 9 × 5 × 5

= 7 × 7 × 3 × 3 × 3 × 3 × 5 × 5

= (7 × 5 × 3 × 3) × (7 × 5 × 3 × 3)

= 315 × 315

= (315)²

Hence,

It is a perfect square.

(iv) 1176

In order to find if the given number is a perfect square,

At first,

We’ll resolve the given number into prime factors:

Hence,

1176 = 7 × 168

= 7 × 8 × 21

= 7 × 2 × 2 × 2 × 7 × 3

Hence,

We can see that,

The number 1176 cannot be expressed as a product of two equal numbers.

Thus,

1176 is not a perfect square.

(v) 5625

In order to find if the given number is a perfect square,

At first,

We’ll resolve the given number into prime factors:

Hence,

5625 = 225 × 25

= 9 × 25 × 25

= 5 × 5 × 5 × 5 × 3 × 3

= (5 × 5 × 3) × (5 × 5 × 3)

= 75 × 75

= (75)²

Hence,

It is a perfect square.

(vi) 9075

In order to find if the given number is a perfect square,

At first,

We’ll resolve the given number into prime factors:

Hence,

9075 = 25 × 363

= 25 × 3 × 121

= 5 × 5 × 3 × 11 × 11

= 25 × 3 × 121

Hence,

We can see that,

The number 9075 cannot be expressed as a product of two equal numbers.

Thus,

9075 is not a perfect square.

(vii) 4225

In order to find if the given number is a perfect square,

At first,

We’ll resolve the given number into prime factors:

Hence,

4225 = 25 × 169

= 5 × 5 × 13 × 13

= (5 × 13) × (5 × 13)

= 65 × 65

= (65)²

Hence,

It is a perfect square.

(viii) 1089

In order to find if the given number is a perfect square,

At first,

We’ll resolve the given number into prime factors:

Hence,

1089 = 121 × 9

= 11 × 11 × 3 × 3

= 11 × 11 × 3 × 3

= (11 × 3) × (11 × 3)

= 33 × 33

= (33)²

Hence,

It is a perfect square.

(i) 1225

In order to show that the given number is a perfect square,

At first,

We’ll resolve the given number into prime factors:

Hence,

1225 = 25 × 49

= 5 × 5 × 7 × 7

= (5 × 7) × (5 × 7)

= 35 × 35

= (35)²

Hence,

The given number is a perfect square.

And,

It is a perfect square of 35.

(ii) 2601

In order to show that the given number is a perfect square,

At first,

We’ll resolve the given number into prime factors:

Hence,

2601 = 9 × 289

= 3 × 3 × 17 × 17

= (3 × 17) × (3 × 17)

= 51 × 51

= (51)²

Hence,

The given number is a perfect square.

And,

It is a perfect square of 51.

(iii) 5929

In order to show that the given number is a perfect square,

At first,

We’ll resolve the given number into prime factors:

Hence,

5929 = 11 × 539

= 11 × 7 × 77

= 11 × 7 × 11 × 7

= (11 × 7) × (11 × 7)

= 77 × 77

= (77)²

Hence,

The given number is a perfect square.

And,

It is a perfect square of 77.

(iv) 7056

In order to show that the given number is a perfect square,

At first,

We’ll resolve the given number into prime factors:

Hence,

7056 = 12 × 588

= 12 × 7 × 84

= 12 × 7 × 12 × 7

= (12 × 7) × (12 × 7)

= 84 × 84

= (84)²

Hence,

The given number is a perfect square.

And,

It is a perfect square of 84.

(v) 8281

In order to show that the given number is a perfect square,

At first,

We’ll resolve the given number into prime factors:

Hence,

8281 = 49 × 169

= 7 × 7 × 13 × 13

= (13 × 7) × (13 × 7)

= 91 × 91

= (91)²

Hence,

The given number is a perfect square.

And,

It is a perfect square of 91.

(i) 3675

At first,

We’ll resolve the given number into prime factors:

Hence,

3675 = 3 × 25 × 49

= 7 × 7 × 3 × 5 × 5

= (5 × 7) × (5 × 7) × 3

In the above factors only 3 is unpaired

So, in order to get a perfect square the given number should be multiplied by 3

Hence,

The number whose perfect square is the new number is as following:

= (5 × 7) × (5 × 7) × 3 × 3

= (5 × 7 × 3) × (5 × 7 × 3)

= (5 × 7 × 3)²

= (105)²

(ii) 2156

At first,

We’ll resolve the given number into prime factors:

Hence,

2156 = 4 × 11 × 49

= 7 × 7 × 2 × 2 × 11

= (2 × 7) × (2 × 7) × 11

In the above factors only 11 is unpaired

So, in order to get a perfect square the given number should be multiplied by 11

Hence,

The number whose perfect square is the new number is as following:

= (2 × 7) × (2 × 7) × 11 × 11

= (2 × 7 × 11) × (2 × 7 × 11)

= (5 × 7 × 11)²

= (154)²

(iii) 3332

At first,

We’ll resolve the given number into prime factors:

Hence,

3332 = 4 × 17 × 49

= 7 × 7 × 2 × 2 × 17

= (2 × 7) × (2 × 7) × 17

In the above factors only 17 is unpaired

So, in order to get a perfect square the given number should be multiplied by 17

Hence,

The number whose perfect square is the new number is as following:

= (2 × 7) × (2 × 7) × 17 × 17

= (2 × 7 × 17) × (2 × 7 × 17)

= (2 × 7 × 17)²

= (238)²

(iv) 2925

At first,

We’ll resolve the given number into prime factors:

Hence,

2925 = 9 × 25 × 13

= 3 × 3 × 13 × 5 × 5

= (5 × 3) × (5 × 3) × 13

In the above factors only 13 is unpaired

So, in order to get a perfect square the given number should be multiplied by 13

Hence,

The number whose perfect square is the new number is as following:

= (5 × 3) × (5 × 3) × 13 × 13

= (5 × 3 × 13) × (5 × 3 × 13)

= (5 × 3 × 13)²

= (195)²

(v) 9075

At first,

We’ll resolve the given number into prime factors:

Hence,

9075 = 3 × 25 × 121

= 11 × 11 × 3 × 5 × 5

= (5 × 11) × (5 × 11) × 3

In the above factors only 3 is unpaired

So, in order to get a perfect square the given number should be multiplied by 3

Hence,

The number whose perfect square is the new number is as following:

= (5 × 11) × (5 × 11) × 3 × 3

= (5 × 11 × 3) × (5 × 11 × 3)

= (5 × 11 × 3)²

= (165)²

(vi) 7623

At first,

We’ll resolve the given number into prime factors:

Hence,

7623 = 9 × 7 × 121

= 7 × 3 × 3 × 11 × 11

= (11 × 3) × (11 × 3) × 7

In the above factors only 7 is unpaired

So, in order to get a perfect square the given number should be multiplied by 7

Hence,

The number whose perfect square is the new number is as following:

= (3 × 11) × (3 × 11) × 7 × 7

= (11 × 7 × 3) × (11 × 7 × 3)

= (11 × 7 × 3)²

= (231)²

(vii) 3380

At first,

We’ll resolve the given number into prime factors:

Hence,

3380 = 4 × 5 × 169

= 2 × 2 × 13 × 13 × 5

= (2 × 13) × (2 × 13) × 5

In the above factors only 5 is unpaired

So, in order to get a perfect square the given number should be multiplied by 5

Hence,

The number whose perfect square is the new number is as following:

= (2 × 13) × (2 × 13) × 5 × 5

= (5 × 2 × 13) × (5 × 2 × 13)

= (5 × 2 × 13)²

= (130)²

(viii) 2475

At first,

We’ll resolve the given number into prime factors:

Hence,

2475 = 11 × 25 × 9

= 11 × 3 × 3 × 5 × 5

= (5 × 3) × (5 × 3) × 11

In the above factors only 11 is unpaired

So, in order to get a perfect square the given number should be multiplied by 11

Hence,

The number whose perfect square is the new number is as following:

=(5 × 3) × (5 × 3) × 11 × 11

= (5 × 11 × 3) × (5 × 11 × 3)

= (5 × 11 × 3)²

= (165)²

(i) 1575

At first,

We’ll resolve the given number into prime factors:

Hence,

1575 = 7 × 25 × 9

= 7 × 3 × 3 × 5 × 5

= (5 × 3) × (5 × 3) × 7

In the above factors only 7 is unpaired

So, in order to get a perfect square the given number should be divided by 7

Hence,

The number whose perfect square is the new number is as following:

= (5 × 3) × (5 × 3)

= (5 × 3) × (5 × 3)

= (5 × 3)²

= (15)²

(ii) 9075

At first,

We’ll resolve the given number into prime factors:

Hence,

9075 = 121 × 25 × 3

= 11 × 11 × 3 × 5 × 5

= (5 × 11) × (5 × 11) × 3

In the above factors only 3 is unpaired

So, in order to get a perfect square the given number should be divided by 3

Hence,

The number whose perfect square is the new number is as following:

=(5 × 11) × (5 × 11)

= (5 × 11)²

= (55)²

(iii) 4851

At first,

We’ll resolve the given number into prime factors:

Hence,

4851 = 11 × 49 × 9

= 11 × 3 × 3 × 7 × 7

= (7 × 3) × (7 × 3) × 11

In the above factors only 11 is unpaired

So, in order to get a perfect square the given number should be divided by 11

Hence,

The number whose perfect square is the new number is as following:

=(7 × 3) × (7 × 3)

= (7 × 3)²

= (21)²

(iv) 3380

At first,

We’ll resolve the given number into prime factors:

Hence,

3380 = 4 × 5 × 169

= 2 × 13 × 13 × 2 × 5

= (2 × 13) × (2 × 13) × 5

In the above factors only 5 is unpaired

So, in order to get a perfect square the given number should be divided by 5

Hence,

The number whose perfect square is the new number is as following:

=(2 × 13) × (2 × 13)

= (2 × 13)²

= (26)²

(v) 4500

At first,

We’ll resolve the given number into prime factors:

Hence,

4500 = 4 × 125 × 9

= 2 × 2 × 3 × 3 × 5 × 5 × 5

= (5 × 3 × 2) × (5 × 3 × 2) × 5

In the above factors only 5 is unpaired

So, in order to get a perfect square the given number should be divided by 5

Hence,

The number whose perfect square is the new number is as following:

=(5 × 3 × 2) × (5 × 3 × 2)

= (5 × 2 × 3) × (5 × 2 × 3)

= (5 × 2 × 3)²

= (30)²

(vi) 7776

At first,

We’ll resolve the given number into prime factors:

Hence,

7776 = 32 × 243

= 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 2

= (2 × 2 × 3 × 3) × (2 × 2 × 3 × 3) × 2 × 3

In the above factors only 2 and 3 are unpaired

So, in order to get a perfect square the given number should be divided by 6

Hence,

The number whose perfect square is the new number is as following:

= (2 × 2 × 3 × 3) × (2 × 2 × 3 × 3)

= (2 × 2 × 3 × 3)²

= (36)²

(vii) 8820

At first,

We’ll resolve the given number into prime factors:

Hence,

8820 = 4 × 5 × 9 × 49

= 2 × 2 × 3 × 3 × 7 × 7 × 5

= (7 × 3 × 2) × (7 × 3 × 2) × 5

In the above factors only 5 is unpaired

So, in order to get a perfect square the given number should be divided by 5

Hence,

The number whose perfect square is the new number is as following:

=(7 × 3 × 2) × (7 × 3 × 2)

= (7 × 3 × 2)²

= (42)²

(viii) 4056

At first,

We’ll resolve the given number into prime factors:

Hence,

4056 = 8 × 3 × 169

= 2 × 2 × 13 × 13 × 3 × 2

= (13 × 2) × (13 × 2) × 6

In the above factors only 6 is unpaired

So, in order to get a perfect square, the given number should be divided by 6

Hence,

The number whose perfect square is the new number is as following:

=(13 × 2) × (13 × 2)

= (13 × 2)²

= (26)²

Let us take the first 3-digit number

First 3-digit number = 100

Now,

We know that,

100 is a perfect square.

And,

Its square root is 10.

Now,

The number before 10 is 9.

Square of 9 = 81

Hence,

The largest 2-digit number which is a perfect square is 81.

At first,

The largest 3 digit number = 999

Now,

The number whose square is 999 is 31.61 (approx.)

Hence,

The square of any number greater than 31.61 would be a 4-digit number.

Therefore,

The square of 31 will be the greatest 3 digit perfect square.

We can calculate the largest 3 digit perfect square number as:

31² = 31 × 31

= 961

(i) We know that,

A number which ends with 2 is not a perfect square

Also, the given number 5372 is ending with the digit 2

Therefore,

The given number is not a perfect square

(ii) We know that,

A number which ends with 3 is not a perfect square

Also, the given number 5963 is ending with the digit 3

Therefore,

The given number is not a perfect square

(iii) We know that,

A number which ends with 7 is not a perfect square

Also, the given number 8457 is ending with the digit 7

Therefore,

The given number is not a perfect square

(iv) We know that,

A number which ends with 8 is not a perfect square

Also, the given number 9468 is ending with the digit 8

Therefore,

The given number is not a perfect square

(v) We know that,

Any number which ends with an odd number of zeros is not a perfect square

Also, the given number 360 is ending with the digit 0

Therefore,

The given number is not a perfect square

(vi) We know that,

Any number which ends with an odd number of zeros is not a perfect square

Also, the given number 6400 is ending with the digit 0

Therefore,

The given number is not a perfect square

(vii) We know that,

Any number which ends with an odd number of zeros is not a perfect square

Also, the given number 2500000 is ending with the digit 0

Therefore,

The given number is not a perfect square

(i) We know that,

The square of an even number is always even

The given number is ending with the digit 6 which is an even number

Thus, it must be a square of even number

(ii) We know that,

The square of an even number is always even

The given number is ending with the digit 1 which is an odd number

Thus, it is not square of even number

(iii) We know that,

The square of an even number is always even

The given number is ending with the digit 0 which is an even number

Thus, it must be a square of even number

(iv) We know that,

The square of an even number is always even

The given number is ending with the digit 5 which is an odd number

Thus, it is not a square of even number

(v) We know that,

The square of an even number is always even

The given number is ending with the digit 4 which is an even number

Thus, it must be a square of even number

(i) We know that,

According to the property of squares, the square of an odd number is an odd number

The given number is ending with the digit 4 which is an even number

Thus, this number is not the square of an odd number.

(ii) We know that,

According to the property of squares, the square of an odd number is an odd number

The given number is ending with the digit 1 which is an odd number

Thus, this number is the square of an odd number.

(iii) We know that,

According to the property of squares, the square of an odd number is an odd number

The given number is ending with the digit 6 which is an even number

Thus, this number is not the square of an odd number.

(iv) We know that,

According to the property of squares, the square of an odd number is an odd number

The given number is ending with the digit 9 which is an odd number

Thus, this number is the square of an odd number

(v) We know that,

According to the property of squares, the square of an odd number is an odd number

The given number is ending with the digit 5 which is an odd number

Thus, this number is the square of an odd number

(i) We know that,

Sum of first n odd numbers = n²

Applying this formula in the question, we get

(1 + 3 + 5 + 7 + 9 + 11 + 13) = (7)²

= 49

(ii) We know that,

Sum of first n odd numbers = n²

Applying this formula in the question, we get

(1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19) = (10)²

= 100

(iii) We know that,

Sum of first n odd numbers = n²

Applying this formula in the question, we get

(1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23) = (12)²

= 144

We know that,

Sum of first n odd numbers = n²

Expressing 81 as a sum of 9 odd numbers

81 = (9)²

n = 9

81 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17

We know that,

Sum of first n odd numbers = n²

Expressing 100 as a sum of 10 odd numbers

100 = (10)²

n = 10

100 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19

(i) As we know that,

For every number m > 1, the Pythagorean triplet is (2m, m² – 1, m² + 1)

Using this result in the question, we get

2m = 6

m = 3

m² = 9

m² – 1 = 9 – 1 = 8

m² + 1 = 9 + 1 = 10

Thus,

The Pythagorean triplet is [6, 8, 10]

(ii) As we know that,

For every number m > 1, the Pythagorean triplet is (2m, m² – 1, m² + 1)

Using this result in the question, we get

2m = 14

m = 7

m² = 49

m² – 1 = 49 – 1 = 48

m² + 1 = 49 + 1 = 50

Thus,

The Pythagorean triplet is [14, 48, 50]

(iii) As we know that,

For every number m > 1, the Pythagorean triplet is (2m, m² – 1, m² + 1)

Using this result in the question, we get

2m = 16

m = 8

m² = 64

m² – 1 = 64 – 1 = 63

m² + 1 = 64 + 1 = 65

Thus,

The Pythagorean triplet is [16, 63, 65]

(iv) As we know that,

For every number m > 1, the Pythagorean triplet is (2m, m² – 1, m² + 1)

Using this result in the question, we get

2m = 20

m = 10

m² = 100

m² – 1 = 100 – 1 = 99

m² + 1 = 100 + 1 = 101

Thus,

The Pythagorean triplet is [20, 99, 101]

(i) We know that,

[(n + 1)² – n²] = (n + 1) + n

(38)² – (37)² = 38 + 37

= 75

(ii) We know that,

[(n + 1)² – n²] = (n + 1) + n

(75)² – (74)² = 75 + 74

= 149

(iii) We know that,

[(n + 1)² – n²] = (n + 1) + n

(92)² – (91)² = 92 + 91

= 183

(iv) We know that,

[(n + 1)² – n²] = (n + 1) + n

(105)² – (104)² = 105 + 104

= 209

(v) We know that,

[(n + 1)² – n²] = (n + 1) + n

(141)² – (140)² = 141 + 140

= 281

(vi) We know that,

[(n + 1)² – n²] = (n + 1) + n

(218)² – (217)² = 218 + 217

= 435

(i) We know that,

(a + b)² = (a² + 2ab + b²)

We have,

310² = (300 + 10)²

= [300² + 2 (300 × 10) + 10²]

= 90000 + 6000 + 100

= 96100

(ii) We know that,

(a + b)² = (a² + 2ab + b²)

We have,

508² = (500 + 8)²

= [500² + 2 (500 × 8) + 8²]

= 250000 + 8000 + 64

= 258064

(iii) We know that,

(a + b)² = (a² + 2ab + b²)

We have,

630² = (600 + 30)²

= [600² + 2 (600 × 30) + 30²]

= 360000 + 36000 + 900

= 396900

(i) We know that,

(a – b)² = (a² – 2ab + b²)

We have

(196)² = (200 - 4)²

= 200² – 2 (200 × 4) + 4²

= 40000 – 1600 + 16

= 3814

(ii) We know that,

(a – b)² = (a² – 2ab + b²)

We have

(689)² = (700 - 11)²

= 700² – 2 (700 × 11) + 11²

= 490000 – 15400 + 121

= 474721

(iii) We know that,

(a – b)² = (a² – 2ab + b²)

We have

(891)² = (900 - 9)²

= 900² – 2 (900 × 9) + 9²

= 810000 – 16200 + 81

= 793881

(i) We have

69 × 71 = (70 – 1) × (70 + 1)

= (70² – 1²)

= 4900 – 1

= 4899

(ii) We have

94 × 106 = (100 – 6) × (100 + 6)

= (100² – 6²)

= 10000 – 36

= 9964

(i) We have

88 × 92 = (90 – 2) × (90 + 2)

= (90² – 2²)

= 8100 – 4

= 8096

(ii) We have

78 × 82 = (80 – 2) × (80 + 2)

= (80² – 2²)

= 6400 – 4

= 6396

(i) The square of an even number is even

(ii) The square of an odd number is odd

(iii) The square of a proper fraction is smaller than the given fraction

(iv) n² = the sum of first n odd natural numbers

(i) The given statement is False

As, the number of digits in a square can also be odd

e.g.: 121

(ii) The given statement is False

As, a prime number is one that is not divisible by any other number except by itself and 1

Thus, square of any number cannot be a prime number

(iii) The given statement is False

Let us take an example:

4 + 9 = 13

As, 4 and 9 are perfect squares of 2 and 3 respectively and their sum i.e., 13 is not a perfect square

(iv) The given statement is also False

Let us take an example:

36 – 25 = 11

As, 36 and 25 are perfect squares and their difference is 11 which is not a perfect square

(v) The given statement is True

Using column method, we get

Therefore,

a = 2

b = 3

Therefore,

23² = 529

Using column method, we get

Therefore,

a = 3

b = 5

Therefore,

35² = 1225

Using column method, we get

Therefore,

a = 5

b = 2

Therefore,

52² = 2704

Using column method, we get

Therefore,

a = 9

b = 6

Therefore,

96² = 9216

Using diagonal method, we get:

Therefore,

67² = 4489

Using diagonal method, we get

Therefore,

86² = 7396

Using diagonal method, we get

Therefore,

137² = 18769

Using diagonal method, we get

Therefore,

256² = 65536

By using prime factorization method, we get

225 = 3 × 3 × 5 × 5

= 3 × 5 = 15

By using prime factorization method, we get

441 = 3 × 3 × 7 × 7

= 3 × 7 = 21

By using prime factorization method, we get

729 = 3 × 3 × 3 × 3 × 3 × 3

= 3 × 3 × 3 = 27

By using prime factorization method, we get

1296 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3

= 2 × 2 × 3 × 3 = 36

By resolving given number into prime factors, we get

2025 = 3 × 3 × 3 × 3 × 5 × 5

Therefore,

= 3 × 3 × 5 = 45

By resolving given number into prime factors, we get

4096 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

Therefore,

= 2 × 2 × 2 × 2 × 2 × 2 = 64

By resolving given number into prime factors, we get

4096 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7

Therefore,

= 2 × 2 × 3 × 7 = 84

By resolving given number into prime factors, we get

4096 = 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5

Therefore,

= 2 × 3 × 3 × 5 = 90

By resolving given number into prime factors, we get

9216 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3

Therefore,

= 2 × 2 × 2 × 2 × 2 × 3 = 96

By resolving given number into prime factors, we get

4096 = 3 × 3 × 5 × 5 × 7 × 7

Therefore,

= 3 × 5 × 7 = 105

By resolving given number into prime factors, we get

4096 = 2 × 2 × 3 × 3 × 3 × 3 × 7 × 7

Therefore,

= 2 × 3 × 3 × 7 = 126

By resolving given number into prime factors, we get

17424 = 2 × 2 × 2 × 2 × 3 × 3 × 11 × 11

Therefore,

= 2 × 2 × 3 × 11 = 132

Resolving 252 into prime factors, we get

252 = 2 × 2 × 3 × 3 × 7

Thus, the 253 must be multiplied by 7 in order to get a perfect square

Therefore,

New number = 252 × 7 = 1764

Hence,

= 2 × 3 × 7

= 42

Resolving 2925 into prime factors, we get

2925 = 3 × 3 × 5 × 5 × 13

Thus, 13 is the smallest number by which 2925 must be divided in order to get a perfect square

Therefore,

New number = = 225

Hence,

= 3 × 5

= 15

Let the number of rows be x

Therefore,

The number of plants in each row is also x

Hence,

Total number of plants = (x × x) = x² = 1225

x² = 1225 = 5 × 5 × 7 × 7

x = = 5 × 7 = 35

Thus,

The total number of rows is 35 and the number of plants in each row is also 35

Let, the number of students be x

Hence,

The amount contributed by each student is Rs x

Total amount contributed = x × x = x² = 1156

1156 = 2 × 2 × 17 × 17

x = = 2 × 17 = 34

Therefore,

The strength of class is 34

We know that,

The smallest number that is divisible by each o0f these numbers is their L.C.M

So,

L.C.M of 6, 9, 15, 20 = 180

Resolving into prime factors, we get

180 = 2 × 2 × 3 × 3 × 5

So, for making it a perfect square we have to multiply it by 5

Multiplying the number by 5, we get

Required number = 180 × 5

= 900

We know that,

The smallest number that is divisible by each o0f these numbers is their L.C.M

So,

L.C.M of 8, 12, 15, 20 = 120

Resolving into prime factors, we get

120 = 2 × 2 × 2 × 3 × 5

So, for making it a perfect square we have to multiply it by 2 × 3 × 5 = 30

Multiplying the number by 30, we get

Required number = 120 × 30

= 3600

According to question,

In order to find the square root of the given number we will use the long division method

Hence,

Using long division method,

Hence,

The square root of number is 24

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 38

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 67

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 79

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 84

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 95

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 107

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 119

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 102

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 134

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 140

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 304

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Therefore, the number that should be subtracted from the given number to make it a perfect square is 9

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Therefore, the number that should be subtracted from the given number to make it a perfect square is 12

Therefore,

Perfect square = 7581 – 12

= 7569

Therefore, its square root is 87

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Therefore, to get a perfect square than the given number we have to take the square of the next natural number of the quotient, i.e. 78

792 = 6241

Therefore,

Number that should be added to the given number to make it a perfect square = 6241 – 6203

= 38

Thus, the perfect square obtained is 6241 and its square root is 79

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

The next natural number that is a perfect square can be obtained by squaring the next natural number of the obtained quotient i.e. 91

Therefore,

(91 + 1)² = (92)² = 8464

Hence,

The number that should be added to the given number to make it a perfect square:

= 8464 – 8400

= 64

Thus, the perfect square obtained is 8464 and its square root is 92

We have,

Smallest number of 4 digits = 1000

Using the long division method, we have

From long division method it is clear that, 1000 is not a perfect square and the obtained square root is between 31 and 32

So, by squaring the next integer we will get the perfect square

(32)² = 1024

Thus, 1024 is the smallest four digit perfect square

As,

= 32

We have,

Greatest five digit number = 99999

By using long division method, we get

From long division method it is clear that 99999 is not a perfect square and the square root obtained is between 316 and 317

Therefore, by squaring the smaller number we will get the perfect square that will be less than 99999

(316)² = 99856

Hence,

99856 is the required perfect square whose square root is 316

Given that,

Area of the square field = 60025 m²

Length of each side of the square field = = 245 m

We know that,

Perimeter of the square = 4 × sides

= 4 × 245

= 980 m

= km

It is also given that, the man is cycling at a speed of 18 km/h

Therefore,

Time =

=

= hr

= sec

= 98 × 2 sec

= 196 sec

= 3 min 16 sec

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 1.3

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 5.8

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 12.5

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 8.7

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 3.14

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 3.17

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 1.04

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 0.54

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 1.732

As,

= 1.732

= 1.73 (Correct up to two decimal places)

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 1.673

Therefore,

= 1.673

= 1.67 (Correct up to two decimal places)

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Hence,

The square root of number is 0.948

Therefore,

= 0.948

= 0.95 (Correct up to two decimal places)

Given that,

Length of rectangle = 13.6 meters

Breadth of rectangle = 3.4 meters

We know that,

Area of rectangle = Length × Breadth

= (13.6 × 3.4)

= 46.24 sq m

Therefore,

Area of the square = 46.25 sq m

Length of each side of the square =

Now, by using long division method we get:

= 6.8

Therefore,

The length of a side of the square is 6.8 meters

We have,

=

We know that,

= 4

And,

= 9

Therefore,

=

=

We have,

Now, using long division method we get:

Also,

Therefore,

=

=

We have,

Now, by using long division method we get:

Also,

Therefore,

=

=

We have,

=

Now, by using long division method we get:

= 25

Also,

= 27

Therefore,

=

=

We have,

=

=

=

=

= 1

We have,

=

=

Now, using long division method we get:

= 37

Also,

=

= 2 × 9

= 18

Therefore,

= =

= 2

We have,

=

=

Now, by using long division method we get:

= 17

Also,

=

= 2 × 5 × 3

= 30

Therefore,

=

= 1

We have,

=

=

=

We have,

=

=

=

=

=

We have,

=

=

= 2 × 7 × 9

= 126

We know that,

As per the properties of square,

All the numbers that end with digits 2, 3, 7 or 8 are not a perfect square

Hence,

Considering the property, we get

The number 5478 is not a perfect square

As the last digit of the number is 8.

Therefore,

Option (C) is the correct option

We know that,

As per the properties of square,

All the numbers that end with digits 2, 3, 7 or 8 are not a perfect square

Hence,

Considering the property, we get

The number 2222 is not a perfect square

As the last digit of the number is 2.

Therefore,

Option (D) is the correct option.

We know that,

As per the properties of square,

All the numbers that end with digits 2, 3, 7 or 8 are not a perfect square

Hence,

Considering the property, we get

The number 1843 is not a perfect square

As the last digit of the number is 3.

Therefore,

Option (A) is the correct option.

We know that,

As per the properties of square,

All the numbers that end with digits 2, 3, 7 or 8 are not a perfect square

Hence,

Considering the property, we get

The number 4787 is not a perfect square

As the last digit of the number is 7.

Therefore,

Option (B) is the correct option.

We know that,

As per the properties of square,

All the numbers that end with odd numbers of zeroes are not a perfect square

Hence,

Considering the property, we get

The number 81000 is not a perfect square

As the number of zeroes of this number is 3.

Therefore,

Option (C) is the correct option.

We know that,

As per the properties of square,

A number which is a perfect square cannot have 2, 3, 7 or 8 as their unit digit.

Hence,

Considering the property, we get

That 8 cannot be the unit digit of a perfect square number

Therefore,

Option (D) is the correct option.

We know that,

Proper fraction is a fraction that is less than 1, where the numerator is less than the denominator.

Hence,

We can observe that,

The square of any proper fraction will be smaller than the original fraction.

Therefore,

Option (B) is the correct option.

We know that,

In the given series,

a = 1, d = 3 - 1 = 2

Sum of n numbers =

=

=

=

= n²

Hence,

Option (C) is the correct option.

We know that,

According to the Pythagorean triplet,

For a natural number m,

(2m, m² – 1, m² + 1) is a Pythagorean triplet.

Hence,

Considering the Pythagorean triplet,

Let m = 4

2m = 8

m² – 1 = 15

m²+ 1 = 17

Thus,

(8, 15, 17) is the Pythagorean triplet.

Hence,

Option (D) is the correct option.

For making 176 a perfect square we have to subtract 7 from it as:

176 – 7 = 169

And, we know that:

= 13

Therefore, option (C) is correct

For making 526 a perfect square we have to add 3 on it as:

526 + 3 = 529

And, we know that:

= 23

Therefore, option (A) is correct

For making 15370 a perfect square we have to add 6 on it as:

15370 + 6 = 15376

And, we know that:

= 124

Therefore, option (B) is correct

By using long division method, we have

Hence, option (D) is correct

By using long division method, we have

Therefore, option (C) is correct

We have,

× =

Also,

= 1.2

Hence, option (B) is correct

We have,

=

=

=

=

=

Therefore, option (C) is correct

We can solve the given question as:

=

=

=

=

=

=

Hence,

Option (B) is the correct option

As we know that,

Square of an even number is always an even number.

Hence,

196 is the square of an even number.

Therefore,

Option (A) is the correct option.

As we know that,

Square of an odd number is always an odd number.

Hence,

1369 is the square of an odd number.

Therefore,

Option (C) is the correct option.

According to question,

In order to find the square root of the given number we will use the long division method.

Hence,

Using long division method,

Therefore,

= 106

We know that,

Greatest five digit number = 99999

Now, by using long division method we get:

From above results it is clear that:

316 < < 317

Therefore,

(316)² = 99856

Hence, 99856 is the least four digit perfect square number having square root 316

We know that,

Least four digit number = 1000

Now, by using long division method we get:

From above results it is clear that:

31 < < 32

Therefore,

(32)² = 1024

Hence, 1024 is the least four digit perfect square number having square root 32

By using long division method we get:

Therefore,

= 0.53

We have,

Now, by using long division method we get:

Therefore,

= 1.732

Hence,

The value of up to 2 decimal places is 1.73

We have,

=

=

=

=

=

We know that,

Any number which is ending with 2, 3, 7 and 8 is not a perfect square

Therefore,

1222 is not a perfect square as it is ending with digit 2

Hence, option (D) is correct

We have,

=

=

=

=

= 1

Therefore, option (C) is correct

We know that,

The square of an even number is always number

Hence, 1764 is the square of an even number as it is ending with the digit 4 which is even

Therefore, option (C) is correct

For making 521 a perfect square, we have to add 8 on it as:

521 + 8 = 529

And we know that,

= 23

Hence, option (D) is correct

For making 178 a perfect square we have to subtract 9 from it as:

178 – 9 = 169

And we know that,

= 13

Therefore, option (C) is correct

We have,

× = ×

=

= 2 × 2 × 3 × 7

= 84

Hence, option (B) is correct

(i) We have,

1 + 3 + 5 + 7 + 9 + 11 + 13

We know that,

Sum of first n odd numbers = n²

Therefore,

1 + 3 + 5 + 7 + 9 + 11 + 13 = (7)²

(ii) By using long division method, we have

Therefore,

= 41

(iii) We know that,

The smallest square number which is exactly divisible by 2, 4 and 6 is 36

Also,

L.C.M of 2, 4b and 6 is 12

Prime factorization of 12 = 2 × 2 × 3

Now, for making it a perfect square we have to multiply it by 3

Therefore,

12 × 3 = 36

(iv) We know that,

A given number is a perfect square having n digits, where n is odd. Then, its square root will have () digits