## RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.1

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.1

**Other Exercises**

- RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.1
- RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.2
- RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.3
- RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.4
- RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.5
- RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.6
- RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.7
- RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.8
- RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.9

**Question 1.**

**Which of the following numbers are perfect squares ?**

**(i)484**

**(ii) 625**

**(iii) 576**

**(iv) 941**

**(v) 961**

**(vi) 2500**

**Solution:**

Grouping the factors in pairs, we have left no factor unpaired

∴ 484 is a perfect square of 22

∴ Grouping the factors in pairs, we have left no factor unpaired

∴ 625 is a perfect square of 25.

Grouping the factors in pairs, we see that no factor is left unpaired

∴ 576 is a perfect square of 24

(iv) 941 has no prime factors

∴ 941 is not a perfect square.

(v) 961 =31 x 31

Grouping the factors in pairs, we see that no factor is left unpaired

∴ 961 is a perfect square of 31

Grouping the factors in pairs, we see that no factor is left impaired

∴ 2500 is a perfect square of 50 .

**Question 2.**

**Show that each of the following* numbers is a perfect square. Also find the number whose square is the given number in each case :**

**(i) 1156**

**(ii) 2025**

**(iii) 14641**

**(iv) 4761**

**Solution:**

Grouping the factors in pairs, we see that no factor is left unpaired

∴ 1156 is a perfect square of 2 x 17 = 34

Grouping the factors in pairs, we see that no factor is left unpaired

2025 is a perfect square of 3 x 3 x 5 =45

Grouping the factors in pairs, we see that no factor is left unpaired

∴ 14641 is a perfect square of 11×11 = 121

Grouping the factors in pairs, we see that no factor is left unpaired

∴ 4761 is a perfect square of 3 x 23 = 69

**Question 3.**

**Find the smallest number by which the given number must be multiplied so that the product is a perfect square.**

**(i) 23805**

**(ii) 12150**

**(iii) 7688**

**Solution:**

Grouping the factors in pairs of equal factors, we see that 5 is left unpaird

∴ In order to complete the pairs, we have to multiply 23805 by 5, then the product will be the perfect square.

Requid smallest number = 5

(ii) 12150 = 2 x 3 x 3×3 x 3×3 x 5×5

Grouping the factors in pairs of equal factors, we see that factors 2 and 3 are left unpaired

∴ In order to complete the pairs, we have to multiply 12150 by 2 x 3 =6 i.e., then the product will be the complete square.

∴ Required smallest number = 6

Grouping the factors in pairs of equal factors, we see that factor 2 is left unpaired

∴ In order to complete the pairs we have to multiply 7688 by 2, then the product will be the complete square

∴ Required smallest number = 2

**Question 4.**

**Find the smallest number by which the given number must be divided so that the resulting number is a perfect square.**

**(i) 14283**

**(ii) 1800**

**(iii) 2904**

**Solution:**

Grouping the factors in pairs of equal factors, we see that factors we see that 3 is left unpaired

Deviding by 3, the quotient will the perfect square.

Grouping the factors in pair of equal factors, we see that 2 is left unpaired.

∴ Dividing by 2, the quotient will be the perfect square.

Grouping the factors in pairs of equal factors, we see that 2 x 3 we left unpaired

∴ Dividing by 2 x 3 = 6, the quotient will be the perfect square.

**Question 5.**

**Which of the following numbers are perfect squares ?**

**11, 12, 16, 32, 36, 50, 64, 79, 81, 111, 121**

**Solution:**

11 is not a perfect square as 11 = 1 x 11

12 is not a perfect square as 12 = 2×2 x 3

16 is a perfect square as 16 = 2×2 x 2×2

32 is not a perfect square as 32 = 2×2 x 2×2 x 2

36 is a perfect square as 36 = 2×2 x 3×3

50 is not a perfect square as 50 = 2 x 5×5

64 is a perfect square as 64 = 2×2 x 2×2 x 2×2

79 is not a perfect square as 79 = 1 x 79

81 is a perfect square as 81 = 3×3 x 3×3

111 is not a perfect square as 111 = 3 x 37

121 is a perfect square as 121 = 11 x 11

Hence 16, 36, 64, 81 and 121 are perfect squares.

**Question 6.**

**Using prime factorization method, find which of the following numbers are perfect squares ?**

**∴ 189,225,2048,343,441,2916,11025,3549**

**Solution:**

Grouping the factors in pairs, we see that are 3 and 7 are left unpaired

∴ 189 is not a perfect square

Grouping the factors in pairs, we see no factor left unpaired

∴ 225 is a perfect square

Grouping the factors in pairs, we see no factor left unpaired

∴ 2048 is a perfect square

Grouping the factors in pairs, we see that one 7 is left unpaired

∴ 343 is not a perfect square.

Grouping the factors in pairs, we see that no factor is left unpaired

∴ 441 is a perfect square.

Grouping the factors in pairs, we see that no factor is left unpaired

∴ 2916 is a perfect square.

Grouping the factors in pairs, we see that no factor is left unpaired

∴ 11025 is a perfect square.

Grouping the factors in pairs, we see that 3, no factor 7 are left unpaired

∴ 3549 is a perfect square.

**Question 7.**

**By what number should each of the following numbers be multiplied to get a perfect square in each case ? Also, find the number whose square is the new number.**

**(i) 8820**

**(ii) 3675**

**(iii) 605**

**(iv) 2880**

**(v) 4056**

**(vi) 3468**

**Solution:**

Grouping the factors in pairs, we see that 5 is left unpaired

∴ By multiplying 8820 by 5, we get the perfect square and square root of product will be

= 2 x 3 x 5 x 7 = 210

Grouping the factors in pairs, we see that 3 is left unpaired

∴ Multiplying 3675 by 3, we get a perfect square and square of the product will be

= 3 x 5 x 7 = 105

Grouping the factors in pairs, we see that 5 is left unpaired

∴ Multiplying 605 by 5, we get a perfect square and square root of the product will be

= 5 x 11 =55

Grouping the factors in pairs, we see that 5 is left unpaired

∴ Multiplying 2880 by 5, we get the perfect square.

Square rooi of product will be = 2 x 2 * 2 – 3 x 5 = 120

Grouping the factors in pairs, we see that 2 and 3 are left unpaired

∴ Multiplying 4056 by 2 x 3 i.e., 6, we get the perfect square.

and square root of the product will be

= 2 x 2 x 3 x 13 = 156

Grouping the factors in pairs, we see that 3 is left unpaired

∴ Multiplying 3468 by 3 we get a perfect square, and square root of the product will be 2 x 3 x 17 = 102

Grouping the factors in pairs, we see that 2 and 3 are left unpaired

∴ Multiplying 7776 by 2 x 3 or 6 We get a perfect square and square root of the product will be

= 2 x 2 x 2 x 3 x 3 x 3 = 216

**Question 8.**

**By what numbers should each of the following be .divided to get a perfect square in each case ? Also find the number whose square is the new number.**

**(i) 16562**

**(ii) 3698**

**(iii) 5103**

**(iv) 3174**

**(v) 1575**

**Solution:**

Grouping the factors in pairs, we see that 2 is left unpaired

∴ Dividing by 2, we get the perfect square and square root of the quotient will be 7 x 13 = 91

Grouping the factors in pairs, we see that 2 is left unpaired,

∴ Dividing 3698 by 2, the quotient is a perfect square

and square of quotient will be = 43

Grouping the factors in pairs, we see that 7 is left unpaired

∴ Dividing 5103 by 7, we get the quotient a perfect square.

and square root of the quotient will be 3 x 3 x 3 = 27

Grouping the factors iq pairs, we see that 2 and 3 are left unpaired

∴ Dividing 3174 by 2 x 3 i.e. 6, the quotient will be a perfect square and square root of the quotient will be = 23

Grouping the factors in pairs, we find that 7 is left unpaired i

∴ Dividing 1575 by 7, the quotient is a perfect square

and square root of the quotient will be = 3 x 5 = 15

**Question 9.**

**Find the greatest number of two digits which is a perfect square.**

**Solution:**

The greatest two digit number = 99 We know, 92 = 81 and 102 = 100 But 99 is in between 81 and 100

∴ 81 is the greatest two digit number which is a perfect square.

**Question 10.**

**Find the least number of three digits which is perfect square.**

**Solution:**

The smallest three digit number =100

We know that 92 = 81, 102 = 100, ll2 = 121

We see that 100 is the least three digit number which is a perfect square.

**Question 11.**

**Find the smallest number by which 4851 must be multiplied so that the product becomes a perfect square.**

**Solution:**

By factorization:

Grouping the factors in pairs, we see that 11 is left unpaired

∴ The least number is 11 by which multiplying 4851, we get a perfect square.

**Question 12.**

**Find the smallest number by which 28812 must be divided so that the quotient becomes a perfect square.**

**Solution:**

By factorization,

Grouping the factors in pairs, we see that 13 is left unpaired

∴ Dividing 28812 by 3, the quotient will be a perfect square.

**Question 13.**

**Find the smallest number by which 1152 must be divided so that it becomes a perfect square. Also find the number whose square is the resulting number.**

**Solution:**

By factorization,

Grouping the factors in pairs, we see that one 2 is left unpaired.

∴ Dividing 1152 by 2, we get the perfect square and square root of the resulting number 576, will be 2 x 2 x 2 x 3 = 24

Hope given RD Sharma Class 8 Solutions Chapter 3 Squares and Square Roots Ex 3.1 are helpful to complete your math homework.

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