Exercise 6.1
1. Find the values of each of the following:
(i) 132
Sol:
132 = 13 × 13
= 169
(ii) 73
Sol:
73 = 7 × 7 × 7
= 343
(iii) 34
Sol:
34 = 3 × 3 × 3 × 3
= 81
2. Find the value of each of the following:
(i) (-7)2
Sol:
Given (-7)2
We know that (-a) even number = Positive number
(-a) odd number = Negative number
We have, (-7)2 = (-7) ×(-7)
= 49
(ii) (-3)4
Sol:
Given (-3)4
We know that (-a) even number = positive
(-a) odd number = Negative number
We have, (-3)4 (-3) ×(-3)×(-3) ×(-3)
= 81
(iii) (-5)5
Sol:
Given (-5)5
We know that (-a) even number = positive number
(-a) odd number = Negative number
We have, (-5)5 = (-5) ×(-5) ×(-5)
= – 3125
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
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
5. Simply:
(i) (-2) × (-3)3
(ii) (-3)2 × (-5)3
(iii) (-2)5 × (-10)2
Solution:
(i) (-2) × (-3)3
(-2) ×(-3)3 = (-2) ×(-3) ×(-3)
= (-2) ×(-27)
= -54
(ii) (-3)2 × (-5)3
(-3) ×(-5)3 = (-3) ×(-3) × (-5) ×(-5)
= 9×(-125)
= -1125
(iii) (-2)5 × (-10)2
(-2)5 (-10)2 = (-2)(-2) (-2) (-10) (-10)
= (-32)100
= -3200
6. Simply:
(i) (3/4)2
(ii) (-2/3)4
(iii) (- 4/5)5
Solution:
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
8. Express each of the following in exponential form
Solution:
(i) (-5)×(-5)×(-5) = +25×(-5)
= -125
= (-5)3
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
10. Express each of the following numbers in exponential form
(i) 512
(ii) 625
(iii) 729
Solution:
(i) 512
512 = 2×2×2×2×2×2×2×2
= 29
(ii) 625
625 = 5×5×5×5
= 25
= 522
= 54
(iii) 729
729 = 3×3×3×3×3×3
= 33 × 33
11. Express each of the following numbers as product powers of their prime factors:
(i) 36
(ii) 675
(iii) 392
Solution:
(i) Given 36
Prime factorization of 36 = 2×2×3×3
= 23×72
(ii) Given 675
Prime factorization of 675 = 3×3×3×5×5
= 33×52
(iii) Given 392
Prime factorization of 392 = 2×2×2×7×7
= 23×72
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 = 2×32 ×52
= 2×32×52
(ii) 2800
Using prime factorization of 2800, we have
2800 = 2×2×2×2×5×5×7
= 24×92×7
(iii) Using Prime factorization of 24000, we have
24000= 2×12000
= 2×6×2000
= 2×2×3×2×2×2×2×5×5×5
= 26×31×53
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:
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
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) Given (-1/2)2×23 × (3/4)2
(-1/2)2×23 × (3/4)2 = 1/4×8×9/16
= 9/8
(ii) Given (-3/5)4 × (4/9)4 × (-15/18)2
(-3/5)4 × (4/9)4 × (-15/18)2= (81/625) × (256/6561) ×(225/324)
= (64/18225)
16. If a = 2 and b= 3, the find the values of each of the following
(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