# Squares and Square Roots| NCERT Solutions| Class 8

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 = 12 =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 = 22 =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 = 92 =81

Hence, unit digit of square of 799 is 1.

(iv) 3853

3853

Unit digit of number = 3

Square of the unit digit 3 = 32= 9

Hence, unit digit of square of 3853 is 9.

(v) 1234

1234

Unit digit of number = 4

Square of the unit digit 4 = 42 =16

Hence, unit digit of square of 1234 is 6.

(vi) 26387

26387

Unit digit of number = 7

Square of the unit digit 7 = 72 =49

Hence, unit digit of square of 26387 is 9.

(vii) 52698

52698

Unit digit of number = 8

Square of the unit digit 8 = 82 =64

Hence, unit digit of square of 52698 is 4.

(viii) 99880

99880

Unit digit of number = 0

Square of the unit digit 0 = 02 =0

Hence, unit digit of square of 99880 is 0.

(ix) 12796

12796

Unit digit of number = 6

Square of the unit digit 6 = 62 =36

Hence, unit digit of square of 12796 is 6.

(x) 55555

55555

Unit digit of number = 5

Square of the unit digit 5= 52 =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-

LCM

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:

Helping Topics

Squares and Square Roots

Worksheet Class 8