# NCERT Solutions for Class 7 Maths

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1.    (i) 26 = 2 × 2 × 2 × 2  × 2 × 2 = 64

(ii) 93= 9  ×  9  ×  9 = 729

(iii) 112 = 11  ×  11 = 121

(iv) 54 = 5  ×  5  ×  5  ×  5 = 625

2.    (i) 6 × 6 × 6 ×  6 = 64

(ii) t × t = t2

(iii) b × b × b × b = b4

(iv) 5 × 5 × 7 × 7 × 7 = 52 × 73

(v) 2 × 2 × a × a = 22 × a2

(vi) a × a × a × c × c × c × c × d = a3 × c4 × d

3.    (i) 512 = 2 × 2 × 2  × 2 × 2 × 2 × 2 × 2 × 2 = 29

(ii) 343 = 7 × 7 × 7 = 73

(iii) 729 = 3 × 3 × 3 × 3 × 3 × 3 = 36

(iv) 3125 = 5 × 5 × 5 × 5 × 5 = 55

4.   (i) 4= 4 × 4 × 4 = 64

3= 3 × 3 × 3 × 3 = 81

Since 64 < 81

Thus, 34 is greater than 43.

(ii) 5= 5 × 5 × 5 = 125

3= 3 × 3 × 3 × 3 × 3 = 243

Since, 125 < 243

Thus, 34 is greater than 53.

(iii) 2= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256

82 = 8 ×  8 = 64

Since, 256 > 64

Thus, 28 is greater than 82.

(iv) 1002 = 100 ×  100 = 10,000

2100 = 2 × 2 × 2 × 2 × 2 × ……… 14 times x………….  x 2 = 16,384 x ……. x 2

Since, 10,000 < 16,384  x ……  X  2

Thus, 2100 is greater than 1002.

(v) 210 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1,024

102 = 10  ×  10 = 100

Since, 1,024 > 100

Thus, 210 > 102

5.    (i) 648 = 23 × 34

(ii) 405 = 5 × 34

(iii) 540 = 22 × 33 × 5

(iv) 3,600 = 24  ×  32  ×  52

6.    (i) 2 × 103 = 2 × 10 × 10 × 10 = 2,000

(ii) 72 × 2= 7 × 7 × 2 × 2 = 196

(iii) 23 × 5 = 2 × 2 × 2 × 5 = 40

(iv) 3 × 4= 3 × 4 × 4 × 4 × 4 = 768

(v) 0 × 10= 0 × 10 × 10 = 0

(vi) 53 × 3= 5 × 5 × 3 × 3 × 3 = 675

(vii) 24 × 3= 2 × 2 × 2 × 2 × 3 × 3 = 144

(viii) 32 × 104 = 3 × 3 × 10 × 10 × 10 × 10 = 90,000

7.    (i) (-4)3 = (-4) × (-4) × (-4)= -64

(ii) (-3) × (-2)3 = (-3) × (-2) × (-2) × (-2) = 24

(iii) (-3)2 × (-5)2 = (-3) × (-3) × (-5) × (-5) = 225

(iv) (-2)3 × (-10)3 = (-2) × (-2) × (-2) × (-10) × (-10) × (-10)

8.    (i) 2.7 × 1012 and 1.5 × 108

On comparing the exponents of base 10,

2.7 × 1012 > 1.5 × 108

(ii) 4 × 1014 and 3 × 1017

On comparing the exponents of base 10,

4 × 1014 < 3 × 1017

9.    (i) 32 × 34 × 34 = 3 (2 + 4 + 8) = 314                [∵ am × an = am + n]

(ii) 615 : 610 = 615-10 = 65                  [∵ am ÷ an = am – n]

(iii) a3 × a= a+ 2 = a5                   [∵ am × an = am + n]

(iv) 7x × 72 = 7x + 2                             [∵ am × an = am + n]

(v) (52)3 ÷ 53 = 52 × 3 53 = 56 ÷ 53            [∵ (am)n = am × n]

= 56 – 3 = 53                                            [∵ am ÷ an = am – n]

(vi) 25 × 55 = (2 × 5)5 = 105                       [∵ am × am = (a × b)m]

(vii) a4 × b4 = (a × b)4                                [∵ am × am = (a × b)m]

(viii) (34)3 = 34 × 3 – 312                              [∵ (am)n = am × n]

(ix) (220 ÷ 215) × 23 = (220 – 15) × 2= 25 × 23         [∵ am ÷ an = am – n]

= 25 + 3 = 2                            [∵ am × am = (a × b)m]

(x) 8÷ 82 = 8t – 2           [∵ am ÷ an = am – n] 11.  (i) 10 × 1011 =10011

L.H.S. 101+11 = 1012     and       R.H.S. (102)11 = 1022

Since, L.H.S. ≠ R.H.S.

Therefore, it is false.

(ii) 23 > 52

L.H.S. 23 = 8 and R.H.S. 52 = 25

Since, L.H.S. is not greater than R.H.S.

Therefore, it is false.

(iii) 23 × 32 = 65

L.H.S. 2× 32 = 8 × 9 = 72 and R.H.S. 65 = 7, 776

Since, L.H.S. ≠ R.H.S.

Therefore, it is false.

(iv) 30 = (1000)0

L.H.S. 30 = 1 and R.H.S. (1000)0 = 1

Since, L.H.S. = R.H.S.

Therefore, it is true.

12.  (i) 108 × 192 = (2× 33) × (2× 3)

= 22+6 × 33-1 = 2× 34

(ii) 270 = 2 × 35 × 5

(iii) 729 × 64 = 3× 26

(iv) 768 = 2× 3

13. 14.  (i) 2,79,404

= 2,00,000 + 70,000 + 9,000 + 400 + 00 + 4

= 2 × 100000 + 7 × 10000+ 9 × 1000 + 4 × 100 + 0 × 10 + 4 × 1

= 2 × 105+7 × 10+ 9 × 10+ 4 × 10+ 0 × 10+ 4 × 100

(ii) 30,06,194

= 30,00,000 + 0 + 0 + 6,000 + 100 + 90 + 4

= 3 × 1000000 + 0 × 100000 + 0 × 10000 + 6 × 1000 + 1 × 100 + 9 × 10 + 4 × 1

= 3 × 106 + 0 × 105 + 0 × 104 + 6 × 103 + 1 × 102 + 9 × 10 + 4 × 100

(iii) 28,06,196

= 20,00,000 + 8,00,000 + 0 + 6,000 + 100 + 90 + 6

= 2 × 1000000 + 8 × 100000 + 0 × 10000 + 6 × 1000 + 1 × 100 + 9 × 10 + 6 × 1

= 2 × 106 + 8 × 105 + 0 × 104 + 6 × 103 + 1 × 102 + 9 × 10 + 6 × 100

(iv) 1,20,719

= 1,00,000 + 20,000 + 0 + 700 + 10 + 9

= 1 × 100000 + 2 × 10000 + 0 × 1000 + 7 × 100 + 1 × 10 + 9 × 1

= 1 × 105 + 2 × 104 + 0 × 103 + 7 × 102 + 1 × 101 + 9 × 109

(v) 20,068 = 20,000 + 00 + 00 + 60 + 8

= 2 × 10000 + 0 × 1000 + 0 × 100 + 6 × 10 + 8 × 1

= 2 × 104 + 0 × 103 + 0 × 102 + 6 × 101 + 8 × 100

15.  (a)  8 × 104 + 6 × 103 + 0 × 102 + 4 × 104 + 5 × 100

= 8 × 10000 + 6 × 1000 + O × 100 + 4 × 10 + 5 × 1

= 80000 + 6000 + 0 + 40 + 5

= 86,045

(b) 4 × 105 + 5 × 103 + 3 × 102 + 2 × 100

= 4 × 100000 + 0 × 10000 + 5 × 1000 + 3 × 100 + 0 × 10 + 2 × 1

= 400000 + 0 + 5000 + 3000 + 0 + 2

= 4,05,302

(c) 3 × 104 + 7 × 102 + 5 × 100

= 3 × 10000 + 0 × 1000 + 7 × 100 + 0 × 10 + 5 × 1

= 30000 + 0 + 700 + 0 + 5

= 30,705

(d) 9 × 105 + 2 × 102 + 3 × 101

= 9 × 100000 + 0 × 10000 + 0 × 1000 + 2 × 100 + 3 × 10 + 0 × 1

= 900000 + 0 + 0 + 200 + 30 + 0

= 9,00,230

16.  (i) 5,00,00,000 = 5 × 1,00,00,000 = 5 × 107

(ii) 70,00,000 = 7 × 10,00,000 = 7 × 106

(iii) 3,18,65,00,000 = 31865 × 100000 = 3.1865 × 10000 × 100000 = 3.1865 × 109

(iv) 3,90,878 = 3.90878 × 100000 = 3.90878 × 105

(v) 39087.8 = 3.90878 × 10000 = 3.90878 × 104

(vi) 3908.78 = 3.90878 × 1000 = 3.90878 × 103

17.  (a) The distance between Earth and Moon = 384,000,000 m

= 384 × 1000000 m = 3.84 × 100 × 1000000 = 3.84 × 108 m

(b) Speed of light in vacuum = 300,000,000 m/s

= 3 × 100000000 m/s

= 3 × 108 m/s

(c) Diameter of the Earth = 1,27,56,000 m

= 12756 × 1000 m = 1.2756 × 10000 × 1000 m

= 1.2756 × 107 m

(d) Diameter of the Sun = 1,400,000,000 m

= 14 × 100,000,000 m = 1.4 × 10 × 100,000,000 m

= 1.4 × 109 m

(e) Average of Stars = 100,000,000,000

= 1 × 100,000,000,000 = 1 × 1011

(f) Years of Universe = 12,000,000,000 years

= 12 × 1000,000,000 years

= 1.2 × 10 × 1000,000,000 years

= 1.2 × 1010 years

(g) Distance of the Sun from the centre of the Milky Way Galaxy

= 300,000,000,000,000,000,000 m

= 3 × 100,000,000,000,000,000,000 m

= 3 × 1029 m

(h) Number of molecules in a drop of water weighing 1.8 gm

= 60,230,000,000,000,000,000,000

= 6023 × 10,000,000,000,000,000,000

= 6.023 × 1000 × 10,000,000,000,000,000,000

= 6.023 × 1022

(i) The Earth has Sea water

= 1,353,000,000 km3

= 1,353 × 1000000 km3

= 1.353 × 1000 × 1000,000 km3

= 1.353 × 109 km3

(j) The population of India

= 1,027,000,000

= 1027 × 1000000

= 1.027 × 1000 × 1000000

= 1.027 × 109.

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