*NCERT Solutions of Chapter: Squares and Square Roots. NCERT Solutions along with worksheets and notes for Class 8.*

**Exercise 6.1**

**(1)** **What will be the unit digit of the squares of the following numbers?**

**(i) 81 (ii) 272 (iii) 799 (iv) 3853 (v) 1234**

**(vi) 26387 (vii)52698 (viii) 99880 (ix) 12796 (x) 55555**

**Ans-**

**(i)** 81

Unit digit of number = 1

Square of the unit digit 1 = 1^{2} =1

Hence, unit digit of the square of 81 is 1.

**(ii)** 272

272

Unit digit of number = 2

Square of the unit digit 2 = 2^{2} =4

Hence, unit digit of the square of 272 is 4.

**(iii)** 799

799

Unit digit of number = 9

Square of the unit digit 9 = 9^{2} =81

Hence, unit digit of square of 799 is 1.

**(iv)** 3853

3853

Unit digit of number = 3

Square of the unit digit 3 = 3^{2}= 9

Hence, unit digit of square of 3853 is 9.

**(v)** 1234

1234

Unit digit of number = 4

Square of the unit digit 4 = 4^{2} =16

Hence, unit digit of square of 1234 is 6.

**(vi)** 26387

26387

Unit digit of number = 7

Square of the unit digit 7 = 7^{2} =49

Hence, unit digit of square of 26387 is 9.

**(vii)** 52698

52698

Unit digit of number = 8

Square of the unit digit 8 = 8^{2} =64

Hence, unit digit of square of 52698 is 4.

**(viii)** 99880

99880

Unit digit of number = 0

Square of the unit digit 0 = 0^{2} =0

Hence, unit digit of square of 99880 is 0.

**(ix)** 12796

12796

Unit digit of number = 6

Square of the unit digit 6 = 6^{2} =36

Hence, unit digit of square of 12796 is 6.

**(x)** 55555

55555

Unit digit of number = 5

Square of the unit digit 5= 5^{2} =25

Hence, unit digit of square of 55555 is 5.

**(2)** **The following numbers are obviously not perfect squares. Give reason.**

**(i) 1057 (ii) 23453 (iii) 7928 (iv) 222222 (v) 64000 (vi) 89722 (vii) 222000**

**Ans-**

**(i)** 1057

1057 is not a perfect square because its unit digit is 7.Numbers have 2, 3, 7 and 8 at unit place is not perfect square.

**(ii) **23453

23453 is not a perfect square because its unit digit is 3.Numbers have 2, 3, 7 and 8 at unit place is not perfect square.

**(iii)** 7928

7928 is not a perfect square because its unit digit is 8.Numbers have 2, 3, 7 and 8 at unit place is not perfect square.

**(iv)** 222222

222222 is not a perfect square because its unit digit is 2.Numbers have 2, 3, 7 and 8 at unit place is not perfect square.

**(v)** 64000

64000 is not a perfect square because it has 0 in odd numbers at the end of the number. A number is a perfect square if it has 0 in even number at the end of the number.

**(vi)** 89722

89722 is not a perfect square because its unit digit is 2.Numbers have 2, 3, 7 and 8 at unit place is not perfect square.

**(vii) **222000

222000 is not a perfect square because it has 0 in odd numbers at the end of the number. A number is a perfect square if it has 0 in even number at the end of the number.

**(2)** **Without doing any calculation, find the numbers which are surely not perfect squares.**

**(i) 153 (ii) 257 (iii) 408 (iv) 441**

**Ans-**

**(i)** 153

153 is not a perfect square because its unit digit is 3.Numbers which have 2, 3, 7 and 8 at unit place is not perfect square.

**(ii)** 257

257 is not a perfect square because its unit digit is 7.Numbers have 2, 3, 7 and 8 at unit place is not perfect square.

**(iii)** 408

408 is not a perfect square because its unit digit is 8.Numbers have 2, 3, 7 and 8 at unit place is not perfect square.

**(iv)** 441

If a number has 0, 1, 4, 5, 6, o r9 at one’s place, then it must be a square number.

441 can be a perfect square because its unit digit is 1.

Hence, 153, 257 and 408 are not perfect square.

**(4) Find the square roots of the following numbers by the Prime Factorisation Method.**

**(i) 729 (ii) 400 (iii) 1764 (iv) 4096 (v) 7744 (vi) 9604 (vii) 5929 (viii) 9216 (ix) 529 (x) 8100**

**Ans-**

**(i)** 729

**(5)For each of the following numbers, find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also find the square root of the square number so obtained.**

**(i) 252 (ii) 180 (iii)1008 (iv)2028 (v) 1458 (vi)768**

**Ans-**

**(i)** 252

**(6)** **For each of the following numbers, find the smallest whole number by which it should be divided so as to get a perfect square. Also find the square root of the square number so obtained.**

**(i) 252 (ii) 2925 (iii) 396 (iv) 2645 (v) 2800 (vi) 1620**

**Ans-**

**(i)** 252

**(9) Find the smallest square number that is divisible by each of the numbers 4, 9 and 10.**

**Ans-**

**(10)** **Find the smallest square number that is divisible by each of the numbers 8, 15 and 20.**

**Ans-**

**Exercise 6.4**

**(1)Find the square root of each of the following numbers by Division method.**

**(i) 2304 (ii) 4489 (iii) 3481 (iv) 529 (v) 3249 (vi) 1369 (vii) 5776 (viii) 7921**

**(ix) 576 (x) 1024 (xi) 3136 (xii) 900**

**Ans-**

**(i)** 2304

Square root of 2304 by division method is showing below:

√2304 = 48

Hence, square root of 2304 is 48.

**(ii)** 4489

Square root of 4489 by division method is showing below:

√4489 = 67

Hence, square root of 4489 is 67.

**(iii)** 3481

Square root of 3481 by division method is showing below:

√3481 = 59

Hence, square root of 3481 is 59.

**(iv) **529

Square root of 529 by division method is showing below:

√529 = 23

Hence, square root of 529 is 23.

**(v)** 3249

Square root of 3249 by division method is showing below:

√3249 = 57

Hence, square root of 3249 is 57.

**(vi) **1369

Square root of 1369 by division method is showing below:

√1369 = 37

Hence, square root of 1369 is 37.

**(vii) **5776

Square root of 5776 by division method is showing below:

√5776 = 76

Hence, square root of 5776 is 76.

**(viii)** 7921

Square root of 7921 by division method is showing below:

√7921 = 89

Hence, square root of 7921 is 89.

**(ix)** 576

Square root of 576 by division method is showing below:

√576 = 24

Hence, square root of 576 is 24.

**(x)** 1024

Square root of 1024 by division method is showing below:

√1024 = 32

Hence, square root of 1024 is 32.

**(xi)** 3136

Square root of 3136 by division method is showing below:

√3136 = 56

Hence, square root of 3136 is 56.

**(xii)** 900

Square root of 900 by division method is showing below:

√900 = 30

Hence, square root of 900 is 30.

**(3)** **Find the square root of the following decimal numbers.**

**(i) 2.56 (ii) 7. 29 (iii) 51.84 (iv) 42.25 (v) 31.36**

**Ans-**

**(i)** 2.56

2.56

Square root of the 2.56 by division method is shown below:

Hence, square root of 2.56 = 1.6

**(ii)** 7. 29

7.29

Square root of the 7.29 by division method is shown below:

Hence, square root of 7.29 = 2.7

**(iii) **51.84

51.84

Square root of the 51.84 by division method is shown below:

Hence, square root of 51.84 = 7.2

**(iv)** 42.25

42.25

Square root of the 42.25 by division method is shown below:

Hence, square root of 42.25 = 6.5

**(v)** 31.36

31.36

Square root of the 31.36 by division method is shown below:

Hence, square root of 31. 36 = 5.6

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

**(i) 402 (ii) 1989 (iii) 3250 (iv) 825 (v) 4000**

**Ans-**

**(i)** 402

402

Square root of the 402 by division method is shown below:

Hence, square root of perfect square 400 is 20.

**(ii)** 1989

1989

Square root of the 1989 by division method is shown below:

Hence, square root of perfect square 1936 is 44.

**(iii)** 3250

3250

Square root of the 3250 by division method is shown below:

Square root of perfect square 3249 = 57

Hence, square root of perfect square 3249 is 57.

**(iv)** 825

825

Square root of the 825 by division method is shown below:

Square root of perfect square 784 = 28

Hence, square root of perfect square 784 is 28.

**(v)** 4000

4000

Square root of the 4000 by division method is shown below:

Square root of perfect square 3969 = 63

Hence, square root of perfect square 3969 is 63.

**(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 (ii) 1750 (iii) 252 (iv) 1825 (v) 6412**

**Ans-**

**(i)** 525

525

Square root of the 525 by division method is shown below:

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

**Ans-**

Number of plants = 1000

Square root of the 1000 by division method is shown below:

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