Table of Contents
Exercise 6.1
Question: 1
Find the values of each of the following:
(i) 132
(ii) 73
(iii) 34
Solution:
(i) 132 = 13 × 13
= 169
(ii) 73 = 7 × 7 × 7
= 343
(iii) 34 = 3 × 3 × 3 × 3
= 81
Question: 2
Find the value of each of the following:
(i) (-7)2
(ii) (-3)4
(iii) (-5)5
Solution:
We know that if ‘a’ is a natural number, then
We have,
(i) (-7)2 = (-7) × (-7)
= 49
(ii) (-3)4 = (-3) × (-3) × (-3) × (-3)
= 81
(iii) (-5)5 = (-5) × (-5) × (-5) × (-5) × (-5)
= -3125
Question: 3
Simply:
(i) 3 × 102
(ii) 22 × 53
(iii) 33 × 52
Solution:
(i) 3 × 102 = 3 × 10 × 10
= 3 × 100
= 300
(ii) 22 × 53 = 2 × 2 × 5 × 5 × 5
= 4 × 125
= 500
(iii) 33 × 52 = 3 × 3 × 3 × 5 × 5
= 27 × 25
= 675
Question: 4
Simply:
(i) 32 × 104
(ii) 24 × 32
(iii) 52 × 34
Solution:
(i) 32 × 104 = 3 × 3 × 10 × 10 × 10 × 10
= 9 × 10000
= 90000
(ii) 24 × 32 = 2 × 2 × 2 × 2 × 3 × 3
= 16 × 9
= 144
(iii) 52 × 34 = 5 × 5 × 3 × 3 × 3 × 3
= 25 × 81
= 2025
Question: 5
Simply:
(i) (-2) × (-3)3
(ii) (-3)2 × (-5)3
(iii) (-2)5 × (-10)2
Solution:
(i) (-2) × (-3)3 = (-2) × (-3) × (-3) × (-3)
= (-2) × (-27)
= 54
(ii) (-3)2 × (-5)3 = (-3) × (-3) × (-5) × (-5) × (-5)
= 9 × (-125)
= -1125
(iii) (-2)5 × (-10)2 = (-2) × (-2) × (-2) × (-2) × (-2) × (-10) × (-10)
= (-32) × 100
= -3200
Question: 6
Simply:
(i) (3/4)2
(ii) (-2/3)4
(iii) (- 4/5)5
Solution:
Question: 7
Identify the greater number in each of the following
(i) 25 or 52
(ii) 34 or 43
(iii) 35 or 53
Solution:
(i) 25 or 52
25 = 2 × 2 × 2 × 2 × 2
= 32
52 = 5 × 5
= 25
Therefore, 25 52
(ii) 34 or 43
= 34 = 3 × 3 × 3 × 3
= 81
= 43 = 4 × 4 × 4
= 64
Therefore, 34 43
(iii) 35 or 53
= 35 = 3 × 3 × 3 × 3 × 3
= 243
= 53 = 5 × 5 × 5
= 125
Therefore, 35 53
Question: 8
Express each of the following in exponential form
(i) (-5) × (-5) × (-5)
Solution:
(i) (-5) × (-5) × (-5) = (-5)3
Question: 9
Express each of the following in exponential form
(i) x × x × x × x × a × a × b × b × b
(ii) (-2) × (-2) × (-2) × (-2) × a × a × a
(iii) (-2/3) × (-2/3) × x × x × x
Solution:
(i) x × x × x × x × a × a × b × b × b = x4a2b3
(ii) (-2) × (-2) × (-2) × (-2) × a × a × a = (-2)4a3
(iii) (-2/3) × (-2/3) × x × x × x = (-2/3)2 x3
Question: 10
Express each of the following numbers in exponential form
(i) 512
(ii) 625
(iii) 729
Solution:
(i) 512 = 29
(iii) 625 = 54
(iii) 729 = 36
Question: 11
Express each of the following numbers as a product of powers of their prime factors
(i) 36
(ii) 675
(iii) 392
Solution:
(i) 36 = 2 × 2 × 3 × 3
= 22 × 32
(ii) 675 = 3 × 3 × 3 × 5 × 5
= 33 × 52
(iii) 392 = 2 × 2 × 2 × 7 × 7
= 23 × 72
Question: 12
Express each of the following numbers as a product of powers of their prime factors
(i) 450
(ii) 2800
(iii) 24000
Solution:
(i) 450 = 2 × 3 × 3 × 5 × 5
= 2 × 32 × 52
(ii) 2800 = 2 × 2 × 2 × 2 × 5 × 5 ×7
= 24 × 52 × 7
(iii) 24000 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 5 × 5 × 5
= 25 × 3 × 53
Question: 13
Express each of the following as a rational number of the form p/q
(i) (3/7)2
(ii) (7/9)3
(iii) (-2/3)4
Solution:
Question: 14
Express each of the following rational numbers in power notation
(i) 49/64
(ii) – 64/125
(iii) -12/16
Solution:
(i) 49/64 = (7/8)2
Because 72 = 49 and 82 = 64
(ii) – 64/125 = (- 4/5)3
Because 43 = 64 and 53 = 125
(iii) – (1/216) = – (1/6)3
Because 13 = 1 and 63 = 216
Question: 15
Find the value of the following
(i) (-1/2)2 × 23 × (3/4)2
(ii) (-3/5)4 × (4/9)4 × (-15/18)2
Solution:
(i) (-1/2)2 × 23 × (3/4)2 = 1/4 × 8 × 9/16
= 9/8
(ii) (-3/5)4 × (4/9)4 × (-15/18)2 = 81/625 × 256/6561 × 225/324 = 64/18225
Question: 16
If a = 2 and b= 3, the find the values of each of the followimg
(i) (a + b)a
(ii) (ab)b
(iii) (b/a)b
(iv) (a/b + b/a)a
Solution:
(i) (a + b)a = (2 + 3)2
= (5)2
= 25
(ii) (ab)b = (2 × 3)3
= (6)3
= 216
(iii) (b/a)b = (3/2)3
= 27/8
(iv) (a/b + b/a)a = (2/3 + 3/2)2
= 169/36
Exercise 6.2
Question: 1
Using laws of exponents, simplify and write the answer in exponential form
(i) 23 × 24 × 25
(ii) 512 ÷ 53
(iii) (72)3
(iv) (32)5 ÷ 34
(v) 37 × 27
(vi) (521 ÷ 513) × 57
Solution:
Question: 2
Simplify and express each of the following in exponential form
(i) ((23)4 × 28) ÷ 212
(ii) (82 × 84) ÷ 83
(iii) (57/52) × 53
Solution:
Question: 3
Simplify and express each of the following in exponential form
Solution:
Question: 4
Write 9 × 9 × 9 × 9 × 9 in exponential form with base 3
Solution:
9 × 9 × 9 × 9 × 9 = (9)5 = (32)5
= 310
Question: 5
Simplify and write each of the following in exponential form
Solution:
Question: 6
Simplify
Solution:
Question: 7
Find the values of n in each of the following
Solution:
Question: 8
Solution:
Exercise 6.3
Question: 1
Express the following numbers in the standard form
(i) 3908.78
(ii) 5, 00, 00, 000
(iii) 3, 18, 65, 00, 000
(iv) 846 × 107
(v) 723 × 109
Solution:
(i) 3908.78 = 3.90878 × 103
Since, the decimal point is moved three places to the left
(ii) 5, 00, 00, 000 = 5, 00, 00, 000.00
= 5 × 107
Since, the decimal point is moved seven places to the left
(iii) 3, 18, 65, 00, 000 = 3, 18, 65, 00, 000.00
= 3.1865 × 109
Since, the decimal point is moved nine places to the left
(iv) 846 × 107 = 8.46 × 102 × 107
= 8.46 × 109
Since, the decimal point is moved two places to the left
(v) 723 × 109 = 7.23 × 102 × 109
= 7.23 × 1011
Since, the decimal point is moved two places to the left
Question: 2
Write the following numbers in the usual form
(i) 4.83 × 107
(ii) 3.21 × 105
(iii) 3.5 × 103
Solution:
(i) 4.83 × 107 = 483 × 10(7–2)
= 483 × 105
= 4, 83, 00, 000
Since, the decimal point is moved two places to the right
(ii) 3.21 × 105 = 321 × 10(5–2)
= 321 × 103
= 3, 21, 000
Since, the decimal point is moved two places to the right
(ii) 3.5 × 103 = 35 × 10(3–1)
= 35 × 102
= 3,500
Since, the decimal point is moved one place to the right
Question: 4
Express the numbers appearing in the following statements in the standard form
(i) The distance between the earth and the moon is 384,000,000 metres.
(ii) Diameter of the earth is 1, 27, 56,000 metres.
(iii) Diameter of the sun is 1, 400, 000, 000 metres.
(iv) The universe is estimated to be about 12,000,000,000 years old.
Solution:
(i) The distance between the earth and the moon is 3.84 × 108 metres.
Since, the decimal point is moved eight places to the left
(ii) Diameter of the earth is 1.2756 × 107 metres.
Since, the decimal point is moved seven places to the left
(ii) Diameter of the sun is 1.4 × 109 metres.
Since, the decimal point is moved nine places to the left
(iv) The universe is estimated to be about 1.2 × 1010 years old.
Since, the decimal point is moved ten places to the left
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