# Class 8 NCERT Solutions- Chapter 6 Squares and Square Roots – Exercise 6.4

### Question 1. Find the square root of each of the following numbers by Division method.

### (i) 2304

### Solution:

: Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar.Step 1Thus we have,

: Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend.Step 2Here, we have 23

Divide and get the remainder.

Here, we get 7

: Bring down the remaining number under the next bar to the right of the remainder.Step 3Here, its 04

So now the new dividend is 704.

: Double the quotient and enter it with a blank on its right.Step 4

: Guess a largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.Step 5In this case 88 × 8 = 704.

So we choose the new digit as 8.

Get the remainder.

: Since the remainder is 0 and no digits are left in the given number.Step 6Hence,

√2304 = 48

### (ii) 4489

**Solution:**

: Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar.Step 1Thus we have,

: Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend.Step 2Here, we have 44

Divide and get the remainder.

Here, we get 8

: Bring down the remaining number under the next bar to the right of the remainder.Step 3Here, its 89

So now the new dividend is 889.

: Double the quotient and enter it with a blank on its right.Step 4

: Guess a largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.Step 5In this case 127 × 7 = 889.

So we choose the new digit as 7.

Get the remainder.

: Since the remainder is 0 and no digits are left in the given number.Step 6Hence,

√4489 = 67

### (iii) 3481

**Solution:**

: Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar.Step 1Thus we have,

: Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend.Step 2Here, we have 34

Divide and get the remainder.

Here, we get 9

: Bring down the remaining number under the next bar to the right of the remainder.Step 3Here, its 81

So now the new dividend is 981.

: Double the quotient and enter it with a blank on its right.Step 4

: Guess a largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.Step 5In this case 109 × 9 = 981.

So we choose the new digit as 9.

Get the remainder.

: Since the remainder is 0 and no digits are left in the given number.Step 6Hence,

√3481 = 59

### (iv) 529

**Solution:**

Step 1Thus we have,

Step 2Here, we have 5

Divide and get the remainder.

Here, we get 1

: Bring down the remaining number under the next bar to the right of the remainder.Step 3Here, its 29

So now the new dividend is 129.

: Double the quotient and enter it with a blank on its right.Step 4

Step 5In this case 23 × 3 = 129.

So we choose the new digit as 3.

Get the remainder.

: Since the remainder is 0 and no digits are left in the given number.Step 6Hence,

√529 = 23

### (v) 3249

**Solution:**

Step 1Thus we have,

Step 2Here, we have 32

Divide and get the remainder.

Here, we get 7

: Bring down the remaining number under the next bar to the right of the remainder.Step 3Here, its 49

So now the new dividend is 749.

: Double the quotient and enter it with a blank on its right.Step 4

Step 5In this case 107 × 7 = 749.

So we choose the new digit as 7.

Get the remainder.

: Since the remainder is 0 and no digits are left in the given number.Step 6Hence, √3249 = 57

### (vi) 1369

**Solution:**

Step 1Thus we have,

Step 2Here, we have 13

Divide and get the remainder.

Here, we get 4

: Bring down the remaining number under the next bar to the right of the remainder.Step 3Here, its 69

So now the new dividend is 469.

: Double the quotient and enter it with a blank on its right.Step 4

Step 5In this case 37 × 7 = 469.

So we choose the new digit as 7.

Get the remainder.

: Since the remainder is 0 and no digits are left in the given number.Step 6Hence,

√1369 = 37

### (vii) 5776

**Solution:**

Step 1Thus we have,

Step 2Here, we have 57

Divide and get the remainder.

Here, we get 8

: Bring down the remaining number under the next bar to the right of the remainder.Step 3Here, its 76

So now the new dividend is 876.

: Double the quotient and enter it with a blank on its right.Step 4

Step 5In this case 146 × 6 = 876.

So we choose the new digit as 6.

Get the remainder.

: Since the remainder is 0 and no digits are left in the given number.Step 6Hence,

√5776 = 76

### (viii) 7921

**Solution:**

Step 1Thus we have,

Step 2Here, we have 79

Divide and get the remainder.

Here, we get 15

: Bring down the remaining number under the next bar to the right of the remainder.Step 3Here, its 21

So now the new dividend is 1521.

: Double the quotient and enter it with a blank on its right.Step 4

Step 5In this case 169 × 9 = 1521.

So we choose the new digit as 9.

Get the remainder.

: Since the remainder is 0 and no digits are left in the given number.Step 6Hence,

√7921 = 89

### (ix) 576

**Solution:**

Step 1Thus we have,

Step 2Here, we have 5

Divide and get the remainder.

Here, we get 1

: Bring down the remaining number under the next bar to the right of the remainder.Step 3Here, its 76

So now the new dividend is 176.

: Double the quotient and enter it with a blank on its right.Step 4

Step 5In this case 44 × 4 = 176.

So we choose the new digit as 4.

Get the remainder.

: Since the remainder is 0 and no digits are left in the given number.Step 6Hence,

√576 = 24

### (x) 1024

**Solution:**

Step 1Thus we have,

Step 2Here, we have 10

Divide and get the remainder.

Here, we get 1

: Bring down the remaining number under the next bar to the right of the remainder.Step 3Here, its 24

So now the new dividend is 124.

: Double the quotient and enter it with a blank on its right.Step 4

Step 5In this case 62 × 2 = 124.

So we choose the new digit as 2.

Get the remainder.

: Since the remainder is 0 and no digits are left in the given number.Step 6Hence,

√1024 = 32

### (xi) 3136

**Solution:**

Step 1Thus we have,

Step 2Here, we have 31

Divide and get the remainder.

Here, we get 6

: Bring down the remaining number under the next bar to the right of the remainder.Step 3Here, its 36

So now the new dividend is 636.

: Double the quotient and enter it with a blank on its right.Step 4

Step 5In this case 106 × 6 = 636.

So we choose the new digit as 6.

Get the remainder.

: Since the remainder is 0 and no digits are left in the given number.Step 6Hence,

√3136 = 56

### (xii) 900

**Solution:**

Step 1Thus we have,

Step 2Here, we have 9

Divide and get the remainder.

Here, we get 0

: Bring down the remaining number under the next bar to the right of the remainder.Step 3Here, its 0

So now the new dividend is 000.

: Double the quotient and enter it with a blank on its right.Step 4

Step 5In this case 60 × 0 = 000.

So we choose the new digit as 0.

Get the remainder.

: Since the remainder is 0 and no digits are left in the given number.Step 6Hence,

√900 = 30

### Question 2. Find the number of digits in the square root of each of the following numbers (without any calculation).

If n is number of digits in a square number then

Number of digits in the square root = if n is

evenand, if n is

odd.

### (i) 64

**Solution:**

Here, n = 2, which is even

So, number of digits in square root is =

=

= 1

### (ii) 144

**Solution:**

Here, n = 3, which is odd

So, number of digits in square root is =

=

= 2

### (iii) 4489

**Solution:**

Here, n = 4, which is even

So, number of digits in square root is =

=

= 2

### (iv) 27225

**Solution:**

Here, n = 5, which is odd

So, number of digits in square root is =

=

= 3

### (v) 390625

**Solution:**

Here, n = 6, which is even

So, number of digits in square root is =

=

= 3

### Question 3. Find the square root of the following decimal numbers.

### (i) 2.56

**Solution:**

To find the square root of a decimal number we put bars on the integral part of the number in the usual manner. And place bars on the decimal part on every pair of digits beginning with the first decimal place.

We get

Since the remainder is 0 and no digits are left in the given number.

Hence,

√2.56 = 1.6

### (ii) 7.29

**Solution:**

To find the square root of a decimal number we put bars on the integral part of the number in the usual manner. And place bars on the decimal part on every pair of digits beginning with the first decimal place.

We get

Since the remainder is 0 and no digits are left in the given number.

Hence,

√7.29 = 2.7

### (iii) 51.84

**Solution:**

To find the square root of a decimal number we put bars on the integral part of the number in the usual manner. And place bars on the decimal part on every pair of digits beginning with the first decimal place.

We get

Since the remainder is 0 and no digits are left in the given number.

Hence,

√51.84 = 7.2

### (iv) 42.25

**Solution:**

We get

Since the remainder is 0 and no digits are left in the given number.

Hence,

√42.25 = 6.5

### (v) 31.36

**Solution:**

We get

Since the remainder is 0 and no digits are left in the given number.

Hence,

√31.36 = 5.6

### Question 4. Find the least number which must be subtracted from each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.

Here, remainder is the least required number to be subtracted from the given number to get a perfect square.

### (i) 402

**Solution: **

By following all the steps for obtaining square root, we get

Here remainder is 2

2 is the least required number to be subtracted from 402 to get a perfect square

New number = 402 – 2 = 400

Thus,

√400 = 20

### (ii) 1989

**Solution:**

By following all the steps for obtaining square root, we get

Here remainder is 53

53 is the least required number to be subtracted from 1989.

New number = 1989 – 53 = 1936

Thus,

√1936 = 44

### (iii) 3250

**Solution:**

By following all the steps for obtaining square root, we get

Here remainder is 1

1 is the least required number to be subtracted from 3250 to get a perfect square.

New number = 3250 – 1 = 3249

Thus,

√3249 = 57

### (iv) 825

**Solution:**

By following all the steps for obtaining square root, we get

Here, the remainder is 41

41 is the least required number which can be subtracted from 825 to get a perfect square.

New number = 825 – 41 = 784

Thus,

√784 = 28

### (v) 4000

**Solution:**

By following all the steps for obtaining square root, we get

Here, the remainder is 31

31 is the least required number which should be subtracted from 4000 to get a perfect square.

New number = 4000 – 31 = 3969

Thus,

√3969 = 63

### Question 5. Find the least number which must be added to each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.

### (i) 525

**Solution:**

By following all the steps for obtaining square root, we get

Here remainder is 41

It represents that 22

^{2}is less than 525.Next number is 23,

Where, 23

^{2}= 529Hence,

the number to be added = 529 – 525 = 4New number = 525+4 = 529

Thus,

√529 = 23

### (ii) 1750

**Solution:**

By following all the steps for obtaining square root, we get

Here the remainder is 69

It represents that 41

^{2}is less than in 1750.The next number is 42

Where, 42

^{2}= 1764Hence,

number to be added to 1750 = 1764 – 1750 = 14New number = 1750 + 14 = 1764

√1764 = 42

### (iii) 252

**Solution:**

By following all the steps for obtaining square root, we get

Here the remainder is 27.

It represents that 15

^{2}is less than 252.The next number is 16

Where,16

^{2}= 256Hence,

number to be added to 252 = 256 – 252 = 4New number = 252 + 4 = 256

and

√256 = 16

### (iv) 1825

**Solution:**

By following all the steps for obtaining square root, we get

The remainder is 61.

It represents that 42

^{2}is less than in 1825.Next number is 43

Where, 43

^{2}= 1849Hence,

number to be added to 1825 = 1849 – 1825 = 24New number = 1825 + 24 = 1849

and

√1849 = 43

### (v) 6412

**Solution:**

By following all the steps for obtaining square root, we get

Here, the remainder is 12.

It represents that 80

^{2}is less than in 6412.The next number is 81

Where, 81

^{2}= 6561Hence,

the number to be added = 6561 – 6412 = 149New number = 6412 + 149 = 6561

and

√6561 = 81

### Question 6. Find the length of the side of a square whose area is 441 m^{2}.

**Solution:**

Let the side of square be x m.

Area of square = x

^{2}According to the given question,

x

^{2}= 441x = √441

Hence,

the side of square is 21 m.

### Question 7. In a right triangle ABC, ∠B = 90°.

### (a) If AB = 6 cm, BC = 8 cm, find AC

**Solution:**

In right triangle ABC

AC

^{2}= AB^{2}+ BC^{2}[By Pythagoras Theorem]AC

^{2}= 6^{2 }+ 8^{2}AC

^{2}= 100AC = √100

AC = 10 cm

### (b) If AC = 13 cm, BC = 5 cm, find AB

**Solution:**

In right triangle ABC

AC^{2}= AB^{2}+ BC^{2}[By Pythagoras Theorem]13

^{2}= AB^{2}+ 5^{2}AB

^{2}= 13^{2}– 5^{2}AB

^{2}= (13+5) (13-5)AB

^{2}= 18 × 8AB

^{2}= 144AB = √144

AB = 12 cm

### Question 8. A gardener has 1000 plants. He wants to plant these in such a way that the number of rows and the number of columns remain same. Find the minimum number of plants he needs more for this.

**Solution:**

Let the number of rows and columns be x.

Total number of plants = x

^{2}x

^{2}= 1000x = √1000

Here the remainder is 39

So the 31

^{2}is less than 1000.Next number is 32

Where, 32

^{2}= 1024Hence the number to be added = 1024 – 1000 = 24

Hence,

the minimum number of plants required by him = 24.

### Question 9. There are 500 children in a school. For a P.T. drill they have to stand in such a manner that the number of rows is equal to number of columns. How many children would be left out in this arrangement.

**Solution:**

Let the number of rows and columns be x.

Total number of plants = x

^{2}x

^{2}= 500x = √500

Here the remainder is 16

New Number 500 – 16 = 484

and, √484 = 22

Thus,

16 students will be left out in this arrangement.