1

Find the value of:
(i) 26
(ii)93
(iii)112
((iv) 54

Answer

(i) 26 = 2 x 2 x 2 x 2 x 2 x 2 = 64
(ii) 93 = 9 x 9 x 9 = 729
(iii) 112 = 11 x 11 = 121
(iv) 54 = 5 x 5 x 5 x 5 = 625

Exercise 13.1 Page Number 252

2

Express the following in exponential form:
(i) 6 x 6 x 6 x 6
(ii) t x t
(iii) b x b x b x b
(iv) 5 x 5 x 7 x 7 x 7
(v) 2 x 2 x a x a
(vi) a x a x a x c x c x c x c x d

Answer

(i) 6 x 6 x 6 x 6 = 64
(ii) t x t = t2
(iii) b x b x b x b = b4
(iv) 5 x 5 x 7 x 7 x 7 = 52 x 73
(v) 2 x 2 x a x a = 22 x a2
(vi) a x a x a x c x c x c x c x d = a3 x c4 x d

Exercise 13.1 Page Number 252

3

Express each of the following numbers using exponential notation:
(i) 512
(ii) 343
(iii) 729
(iv) 3125

Answer

(i) 512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 29
(ii) 343 = 7 x 7 x 7 = 73
(iii) 729 = 3 x 3 x 3 x 3 x 3 x 3 = 36
(iv) 3125 = 5 x 5 x 5 x 5 x 5 = 55

Exercise 13.1 Page Number 253

4

Identify the greater number, wherever possible, in each of the following:
(i) 43 or 34
(ii) 53 or 35
(iii) 28 or 82
(iv) 1002 or 2100
(v) 210 or 102

Answer

(i) 43= 4 x 4 x 4 = 64
34= 3 x 3 x 3 x 3 = 81
Since 64 < 81
Thus, 34 is greater than 43.

(ii) 53 = 5 x 5 x 5 = 125
35= 3 x 3 x 3 x 3 x 3 = 243
Since, 125 < 243
Thus, 35 is greater than 53.

(iii) 28= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256
8282= 8 x 8 = 64
Since, 256 > 64
Thus, 28 is greater than 82.

(iv) 1002= 100 x 100 = 10,000
2100= 2 x 2 x 2 x 2 x 2 x …..14 times x ……… x 2 = 16,384 x ….. x 2
Since, 10,000 < 16,384 x ……. X 2
Thus, 2100 is greater than1002.

(v) 210= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 1,024
102= 10 x 10 = 100
Since, 1,024 > 100
Thus, 210> 102

Exercise 13.1 Page Number 253

5

Express each of the following as product of powers of their prime factors:
(i) 648
(ii) 405
(iii) 540
(iv) 3,600

Answer

(i) 648 = 23×34
(ii) 405 = 5×34
(iii) 540 = 22×33×5
(iv) 3,600 = 24×32×52

Exercise 13.1 Page Number 253

6

Simplify:
(i) 2×103
(ii) 72×22
(iii)23×5
(iv) 3×44
(v)0×102
(vi) 52 ×33
(vii)24×632
(viii) 32×104

Answer

(i) 2×103= 2 x 10 x 10 x 10 = 2,000

(ii) 72×22= 7 x 7 x 2 x 2 = 196

(iii) 23×5= 2 x 2 x 2 x 5 = 40

(iv) 3×44= 3 x 4 x 4 x 4 x 4 = 768

(v) 0×102= 0 x 10 x 10 = 0

(vi) 52×33= 5 x 5 x 3 x 3 x 3 = 675

(vii) 24×632= 2 x 2 x 2 x 2 x 3 x 3 = 144

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

Exercise 13.1 Page Number 253

7

Simplify:
(i) (-4)3
(ii) (−3)×(−2)3
(iii) (−3)2×(−5)2
(iv) (−2)3 ×(−10)3

Answer

(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)=8000

Exercise 13.1 Page Number 253

8

Compare the following numbers:
(i) 2.7×1012 ;1.5 x 10 8
(ii) 4×1014; 3×1017

Answer

(i) 2.7×1012 and 1.5 x 10 8

On comparing the exponents of base 10,

2.7×1012> 1.5 x 10 8

(ii) 4×1014 and 3×1017

On comparing the exponents of base 10,

4×1014< 3×1017

Exercise 13.1 Page Number 253

1

Using laws of exponents, simplify and write the answer in exponential form:
(i)32×34×38
(ii) 615÷610
(iii) a3×a2
(iv) 7x×72
(v) (52)2÷53
(vi) 25×55
(vii) a4×b4
(viii) (34)3
(ix) (220÷215) x23
(x) 8t÷82

Answer

(i) 32×34×38=3(2+4+8)=314 [∵ am×an=am+n]

(ii) 615÷610=615−10=65 [∵ am÷an=am−n]

(iii) a3×a2=a3+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]= 5

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

(vii) a4×b4=(a×b)4 [∵ am×bm=(a×b)m]

(viii) (34)3= 34×3=312 [∵ (am)n=am×n]

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

= 25 x 23 = 25+3 [∵ am×bm=(a×b)m] = 28

(x) 8t÷82=8t−2 [∵ am÷an=am−n]

Exercise 13.2 Page Number 260

2

Simplify and express each of the following in exponential form:
(i) (23×34×4)/3×32
(ii) [(52)3×54]÷57
(iii) 254÷53
(iv) 3×72×118/21×11
(v) 37/(34×33)
(vi) 20+30+40
(vii) 20×30×40
(viii) (30+20)×50
(ix) 28×a5/43×a3
(x) (a5/a3)×a8
(xi) 45×a8b3/45×a5b2
(xii) (23×2)2

Answer

(i) (23×34×4)/3×32=23×34×22/3×25=23+2×34/3×25 [∵ am×an=am+n]

= 25×34/3×25=25−5×34−3 [∵ am÷an=am−n]

= 20×33= 1×33=33

(ii) [(52)3×54]÷57=[56×54]÷57 [∵ (am)n=am×n]

= [56+4]÷57=510÷57 [∵ am×an=am+n]

= 510−7=53 [∵ am÷an=am−n]

(iii) 254÷53=(52)4÷53=58÷53 [∵ (am)n=am×n]

= 58-3 = 53 [∵ am÷an=am−n]

(iv) 3×72×118/21×113=3×72×118/3×7×113=31−1×72−1×118−3 [∵ am÷an=am−n]

= 30×71×115= 7×115

(v) 37/34×33=37/34+3=37/37 [∵ am×an=am+n]

= 37−7=30 =1 [∵ am÷an=am−n]

(vi) 20+30+40=1+1+1=3 [∵ a0=1]

(vii) 20×30×40=1×1×1=1 [∵ a0=1]

(viii) (30+20)×50=(1+1)×1=2×1=2 [∵ a0=1]

(ix) 28×a5/43×a3=28×a5/(22)3×a3=28×a5/26×a3 [∵ (am)n=am×n]

= 28-6 x a5-3 = 22 x a2

(x) (a5/a3)×a8=(a5−3)×a8=a2×a8 [∵ am÷an=am−n]

= a2+8 = a10 [∵ am×an=am+n]

(xi) 45×a8b3/45×a5b2=45−5×a8−5×b3−2=40×a3×b [∵ am÷an=am−n]

= 1x a3 x b = a3 x b [∵ a0=1]

(xii) (23×2)2=(23+1)2=(24)2 [∵ am×an=am+n]

= 24×2=28 [∵ (am)n=am×n]

Exercise 13.2 Page Number 260

3

Say true or false and justify your answer:
(i) 10×1011=10011
(ii) 23> 52
(iii)23×32=65
(iv) 30= (1000)0

Answer

(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. 23×32=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.

Exercise 13.2 Page Number 260

4

Express each of the following as a product of prime factors only in exponential form:
(i) 108 x 192
(ii) 270
(iii) 729 x 64
(iv) 768

Answer

(i) 108 x 192

= (22×33)×(26×3)

= 22+6×33+1

= 28×34

(ii) 270

= 2×33×5

(iii) 729 x 64

= 36×26

(iv) 768

= 28×3

Exercise 13.2 Page Number 261

5

Simplify:
(i) (25)2×73/83×7
(ii) 25×52×t8/103×t4
(iii) 35×105×25/57×65

Answer

(i) (25)2×73/83×7=25×2×73/(23)3×7

= 210×73/29×7

= 210−9×73−1=2×72

= 2 x 49

= 98

(ii) 25×52×t8/103×t4=52×52×t8/(5×2)3×t4

= 52+2×t8−4/23×53

= 54×t4/23×53

= 54−3×t4/23

= 5t4/8

(iii) 35×105×25/57×65= 35×(2×5)5×52/57×(2×3)5

= 35×25×55×52/57×25×35

= 35×25×55+2/57×25×35

= 35×25×57/57×25×35

= 25−5×35−5×57−7

= 20×30×50

= 1 x 1 x 1

= 1

Exercise 13.2 Page Number 261

1

Write the following numbers in the expanded form:
279404, 3006194, 2806196, 120719, 20068

Answer

(i) 2,79,404 = 2,00,000 + 70,000 + 9,000 + 400 + 00 + 4
= 2 x 100000 + 7 x 10000 + 9 x 1000 + 4 x 100 + 0 x 10 + 4 x 1
= 2 x 105 + 7 x 104 + 9 x 103 + 4 x 102 + 0 x 101 + 4 x 100

(ii) 30,06,194 = 30,00,000 + 0 + 0 + 6,000 + 100 + 90 + 4
= 3 x 1000000 + 0 x 100000 + 0 x 10000 + 6 x 1000 + 1 x 100 + 9 x 10 + 4 x 1
= 3 x 106 + 0 x105 + 0 x 104 + 6 x 103 + 1 x 102 + 9 x 101 + 4 x 100

(iii) 28,06,196 = 20,00,000 + 8,00,000 + 0 + 6,000 + 100 + 90 + 6
= 2 x 1000000 + 8 x 100000 + 0 x 10000 + 6 x 1000 + 1 x 100 + 9 x 10 + 6 x 1
= 2 x 106 + 8 x105 + 0 x 104 + 6 x 103 + 1 x 102 + 9 x 101 + 6 x 100

(iv) 1,20,719 = 1,00,000 + 20,000 + 0 + 700 + 10 + 9
= 1 x 100000 + 2 x 10000 + 0 x 1000 + 7 x 100 + 1 x 10 + 9 x 1
= 1 x105 + 2 x 104 + 0 x 103 + 7 x 102 + 1 x 101 + 9 x 100

(v) 20,068 = 20,000 + 00 + 00 + 60 + 8
= 2 x 10000 + 0 x 1000 + 0 x 100 + 6 x 10 + 8 x 1
= 2 x 104 + 0 x 103 + 0 x 102 + 6 x 101 + 8 x 100

Exercise 13.3 Page Number 263

2

Find the number from each of the following expanded forms:
(a) 8 x 104 + 6 x 103 + 0 x 102 + 4 x 101 + 5 x 100
(b) 4 x105 + 5 x 103 + 3 x 102 + 2 x 100
(c) 3 x 104 + 7 x 102 + 5 x 100
(d) 9 x105 + 2 x 102 + 3 x 101

Answer

(a) 8 x 104 + 6 x 103 + 0 x 102 + 4 x 101 + 5 x 100
= 8 x 10000 + 6 x 1000 + 0 x 100 + 4 x 10 + 5 x 1
= 80000 + 6000 + 0 + 40 + 5 = 86,045

(b) 4 x105 + 5 x 103 + 3 x 102 + 2 x 100
= 4 x 100000 + 5 x 1000 + 3 x 100 + 2 x 1
= 400000 + 5000 + 300 + 2 = 4,05,302

(c) 3 x 104 + 7 x 102 + 5 x 100
= 3 x 10000 + 7 x 100 + 5 x 1
= 30000 + 700 + 5 = 30,705

(d) 9 x105 + 2 x 102 + 3 x 101
= 9 x 100000 + 2 x 100 + 3 x 10
= 900000 + 200 + 30 = 9,00,230

Exercise 13.3 Page Number 263

3

Express the following numbers in standard form:
(i) 5,00,00,000
(ii) 70,00,000
(iii) 3,18,65,00,000
(iv) 3,90,878
(v) 39087.8
(vi) 3908.78

Answer

(i) 5,00,00,000 = 5 x 1,00,00,000 = 5 x 107

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

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

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

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

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

Exercise 13.3 Page Number 263

4

Express the number appearing in the following statements in standard form:
(a) The distance between Earth and Moon is 384,000,000 m.
(b) Speed of light in vacuum is 300,000,000 m/s.
(c) Diameter of Earth is 1,27,56,000 m.
(d) Diameter of the Sun is 1,400,000,000 m.
(e) In a galaxy there are on an average 100,000,000,0000 stars.
(f) The universe is estimated to be about 12,000,000,000 years old.
(g) The distance of the Sun from the centre of the Milky Way Galaxy is estimated to be 300,000,000,000,000,000,000 m.
(h) 60,230,000,000,000,000,000,000 molecules are contained in a drop of water weighing 1.8 gm.
(i) The Earth has 1,353,000,000 cubic km of sea water.
(j) The population of India was about 1,027,000,000 in March, 2001.

Answer

(a) The distance between Earth and Moon = 384,000,000 m
= 384 x 1000000 m = 3.84 x 100 x 1000000 = $3.84 \times 108 " role="presentation" id="MathJax-Element-437-Frame" style="-webkit-tap-highlight-color: transparent; overflow-wrap: normal; max-width: none; max-height: none; min-width: 0px; min-height: 0px; float: none; word-spacing: normal; user-select: auto !important;"> 3.84 \times 10$ 8 $m$

(b) Speed of light in vacuum = 300,000,000 m/s
= 3 x 100000000 m/s = $3 \times 108 " role="presentation" id="MathJax-Element-438-Frame" style="-webkit-tap-highlight-color: transparent; overflow-wrap: normal; max-width: none; max-height: none; min-width: 0px; min-height: 0px; float: none; word-spacing: normal; user-select: auto !important;"> 3 \times 10$ 8 $3 \times 108 " role="presentation" style="-webkit-tap-highlight-color: transparent; overflow-wrap: normal; max-width: none; max-height: none; min-width: 0px; min-height: 0px; float: none; word-spacing: normal; user-select: text !important;">$ m/s

(c) Diameter of the Earth = 1,27,56,000 m
= 12756 x 1000 m = 1.2756 x 10000 x 1000 m = $1.2756 \times 107 " role="presentation" id="MathJax-Element-439-Frame" style="-webkit-tap-highlight-color: transparent; overflow-wrap: normal; max-width: none; max-height: none; min-width: 0px; min-height: 0px; float: none; word-spacing: normal; user-select: auto !important;"> 1.2756 \times 10$ 7 $m$

(d) Diameter of the Sun = 1,400,000,000 m
= 14 x 100,000,000 m = 1.4 x 10 x 100,000,000 m = $1.4 \times 109 " role="presentation" id="MathJax-Element-440-Frame" style="-webkit-tap-highlight-color: transparent; overflow-wrap: normal; max-width: none; max-height: none; min-width: 0px; min-height: 0px; float: none; word-spacing: normal; user-select: auto !important;"> 1.4 \times 10$ 9 $m$

(e) Average of Stars = 100,000,000,000
= 1 x 100,000,000,000 = $1 \times 1011 " role="presentation" id="MathJax-Element-441-Frame" style="-webkit-tap-highlight-color: transparent; overflow-wrap: normal; max-width: none; max-height: none; min-width: 0px; min-height: 0px; float: none; word-spacing: normal; user-select: auto !important;"> 1 \times 10$ 11

(f) Years of Universe = 12,000,000,000 years
= 12 x 1000,000,000 years
= 1.2 x 10 x 1000,000,000 years = $1.2 \times 1010 " role="presentation" id="MathJax-Element-442-Frame" style="-webkit-tap-highlight-color: transparent; overflow-wrap: normal; max-width: none; max-height: none; min-width: 0px; min-height: 0px; float: none; word-spacing: normal; user-select: auto !important;"> 1.2 \times 10$ 10 $years$

(g) Distance of the Sun from the = 300,000,000,000,000,000,000 m
centre of the Milky Way Galaxy = 3 x 100,000,000,000,000,000,000 m
= $3 \times 1020 " role="presentation" id="MathJax-Element-443-Frame" style="-webkit-tap-highlight-color: transparent; overflow-wrap: normal; max-width: none; max-height: none; min-width: 0px; min-height: 0px; float: none; word-spacing: normal; user-select: auto !important;"> 3 \times 10$ 20 $m$

(h) Number of molecules in a drop = 60,230,000,000,000,000,000,000
of water weighing 1.8 gm = 6023 x 10,000,000,000,000,000,000
= 6.023 x 1000 x 10,000,000,000,000,000,000 = $6.023 \times 1022 " role="presentation" id="MathJax-Element-444-Frame" style="-webkit-tap-highlight-color: transparent; overflow-wrap: normal; max-width: none; max-height: none; min-width: 0px; min-height: 0px; float: none; word-spacing: normal; user-select: auto !important;"> 6.023 \times 10$ 22

(i) The Earth has Sea water = 1,353,000,000

Exercise 13.3 Page Number 263

Exponents and Powers

NCERT Solutions for Chapter 13 Exponents and Powers Class 7 Maths

1

2

3

4

5

6

7

8

1

2

3

4

5

1

2

3

4

Exponents and Powers

NCERT Solutions for Chapter 13 Exponents and Powers Class 7 Maths

1

2

3

4

5

6

7

8

1

2

3

4

5

1

2

3

4

Related Chapters