8. POWERS AND EXPONTS

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8. POWERS AND EXPONTS

EX-4.1

Question 1.

Simplify and give reasons
(i) 4-3
(ii) (-2) 7
(iii) (34)−3
(iv) (-3)-4
Solution:
(i) 4-3 143=164 [∵ a-n = 1a′′

(ii) (-2) 7 = -(2) 7 = -128

[∵ 7 is an odd number]
[∵ (-a)n = -(an) if ‘n’ is odd]

(iii) (34)−3 = 3−34−3=4333=(43)3

(iv) (-3)-4 = 1(−3)4 [∵a-n = 1an
= 1(3)4 [∵ 4 is even]
= 181

Question 2.
Simplify the following:
(i) (12)4×(12)5×(12)6
(ii) (-2)7 x (-2)3 x (-2)4
(iii) 44 x (54)4
(iv) (5−45−6) x 53
(v) (-3) 4 x 74
Solution:
(i) (12)4×(12)5×(12)6
(12)4+5+6=(12)15=1215
[∵ am x an = am + n]

(ii) (-2)7 x (-2)3 x (-2)4
(-2)7 + 3 + 4 = (-2) 14 = 2 14
[∵ (-a)n = an is even]

(iii) 44 x (54)4
44 x (54)4 = 54
[ ∵ (ab)m=ambm ]

(iv) (5−45−6) x 53
5-4 x (56 x 53) [ ∵ 1a−n=an
= 5-4 x 56+3 [ ∵ am x an = (a)m+n
= 5-4 x 59
= 5(-4)+9 = 55

(v) (-3) 4 x 74
= 34 x 74[.4isevennumber]
=(3 x 7)4 [:amxbm=(ab)m]
= (21)4

Question 3.
Simplify
(i) 22×322−2×3−1
(ii) (4-1 x 3-1) ÷ 6-1
Solution:
(i) 22×322−2×3−1
= 22 x 22 x 32 x 3-1
= 22+2 x 32 + ( – 1)
=24 x 31 = 16 x 3 = 48

(ii) (4-1 x 3-1) ÷ 6-1
estion 4.

Simplify and give reasons
(i) (40 + 5-1) x 52 x 13
Solution:

(ii) (12)−3×(14)−3×(15)−3
Solution:
= (12×14×15)−3
= (140)−3 [∵ am x bm x cm = (abc)m
= (40)3 [ ∵]1a−n = an ]

(iii) (2-1 + 3-1 + 4-1) x 34
Solution:

(iv) 3−23 x (30 – 3-1
Solution:

(v) 1 + 2-1 + 3-1 + 40
Solution:

(vi) [(32)−2]2
Solution:

Question 5.Simplify and give reasons
(i) [(32−22)÷15]2
Solution:
[(9−4)÷15)]
= [5×51]2 = (52)2 54 = 625 [∵ (am)n = amn]

(ii) ((52)3 x 54) ÷ 56
Solution:

Question 6.
Find the value of ’n’ in each of the following:
(i) (23)3×(23)5=(23)n−2
Solution:

Here bases are equal, so exponents are
also equal.
⇒ n – 2 = 8
⇒ n = 8 + 2 = 10
∴ n = 10

(ii) (-3)n+1 x (-3)5 = (-3)3
Solution:
⇒(-3)n+1+5 = (-3)-4 [∵ am x an = am+n ]
⇒ (-3)n+6 = (-3)-4
⇒ n + 6 = -4
⇒ n = -4 – 6 = -10
⇒ n = -10

(iii) 72n+1 ÷ 49 = 73
Solution:

⇒ 72n+1-2 = 73 [ ∵ aman=am−n ]
⇒ 72n – 1= 73
⇒ 2n – 1 = 3
⇒ 2n = 3 + 1 = 4
⇒ n = 42
∴ n = 2

Question 7.
Find ’x’ if 2-3 = 12x
Solution:
2-3 = 12x = 2-x
⇒ 2-3 = 2-x [ 1an = a-n ]
⇒ -x = -3
∴ x = 3

Question 8.
Simplify [(34)−2÷(45)−3]×(35)−2
Solution:

Question 9.
If m = 3 and n = 2 find the value of
(i) 9m2 – 10n3
(ii) 2m2 n2
(iii) 2m3 + 3n2 – 5m2n
(iv) mn – nm
Solution:
1) 9m2 – 10n3
= 9(3)2 – 10(2)3
= 9 x 9 – 10 x8
= 81 – 80 = 1

(ii) 2m2 n2
= 2(3)2 (2)2
= 2 x 9 x 4 = 72

(iii) 2m3 + 3n2 – 5m2n
= 2(3)3 + 3(2)2 – 5(3)2(2)
= (2 x 27) + (3 x 4) – (5 x 9 x 2)
= 54 + 12 – 90
= 66 – 90 = – 24

(iv) mn – nm
= 32 – 23
= 3 x 3 – 2 x 2 x 2
= 9 – 8 = 1

Question 10.
Simplify and give reasons (47)−5×(74)−7
Solution: