Chapter 6: Exponents 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