Solved Examples on Exponents



Here are some solved examples on exponents using the laws of exponents.

1. Evaluate the exponent:


(i) 5-3

(ii) (1/3)-4

(iii) (5/2)-3

(iv) (-2)-5

(v) (-3/4)-4

We have:

(i) 5-3 = 1/53 = 1/125

(ii) (1/3)-4 = (3/1)4 = 34 = 81

(iii) (5/2)-3 = (2/5)3 = 23/53 = 8/125

(iv) (-2)-5 = 1/(-2)-5 = 1/-25 = 1/-32 = -1/32

(v) (-3/4)-4 = (4/-3)4 = (-4/3)4 = (-4)4/34 = 44/34 = 256/81



2. Evaluate: (-2/7)-4 × (-5/7)2

Solution:

(-2/7)-4 × (-5/7)2

= (7/-2)4 × (-5/7)2

= (-7/2)4 × (-5/7)2 [Since, (7/-2) = (-7/2)]

= (-7)4/24 × (-5)2/72

= {74 × (-5)2}/{24 × 72 } [Since, (-7)4 = 74]

= {72 × (-5)2 }/24

= [49 × (-5) × (-5)]/16

= 1225/16



3. Evaluate: (-1/4)-3 × (-1/4)-2

Solution:


(-1/4)-3 × (-1/4)-2

= (4/-1)3 × (4/-1)2

= (-4)3 × (-4)2

= (-4)(3 + 2)

= (-4)5

= -45

= -1024.



4. Evaluate: {[(-3)/2]2}-3

Solution:


{[(-3)/2]2}-3

= (-3/2)2 × (-3)

= (-3/2)-6

= (2/-3)6

= (-2/3)6

= (-2)6/36

= 26/36

= 64/729



5. Simplify:

(i) (2-1 × 5-1)-1 ÷ 4-1

(ii) (4-1 + 8-1) ÷ (2/3)-1

Solution:

(i) (2-1 × 5-1)-1 ÷ 4-1

= (1/2 × 1/5)-1 ÷ (4/1)-1

= (1/10)-1 ÷ (1/4)

= 10/1 ÷ 1/4

= (10 ÷ 1/4)

= (10 × 4)

= 40.


(ii) (4-1 + 8-1) ÷ (2/3)-1

= (1/4 + 1/8) ÷ (3/2)

= (2 + 1)/8 ÷ 3/2

= (3/8 ÷ 3/2)

= (3/8 ÷ 2/3)

= 1/4



6. Simplify: (1/2)-2 + (1/3)-2 + (1/4)-2

Solution:


(1/2)-2 + (1/3)-2 + (1/4)-2

= (2/1)2 + (3/1)2 + (4/1)2

= (22 + 32 + 42)

= (4 + 9 + 16)

= 29.



7. By what number should (1/2)-1 be multiplied so that the product is (-5/4)-1?

Solution:


Let the required number be x. Then,

x × (1/2)-1 = (-5/4)-1

⇒ x × (2/1) = (4/-5)

⇒ 2x = -4/5

⇒ x = (1/2 × -4/5) = -2/5

Hence, the required number is -2/5.



8. By what number should (-3/2)-3 be divided so that the quotient is (9/4)-2?

Solution:


Let the required number be x. Then,

(-3/2)-3/x = (9/4)-2

⇒ (-2/3)3 = (4/9)2 × x

⇒ (-2)3/33 = 42/92 × x

⇒ -8/27 = 16/81 × x

⇒ x = {-8/27 × 81/16}

⇒ x = -3/2

Hence, the required number is -3/2



9. If a = (2/5)2 ÷ (9/5)0 find the value of a-3.

Solution:


a-3 = [(2/5)2 ÷ (9/5)0]-3

= [(2/5)2 ÷ 1]-3

= [(2/5)2]-3

= (2/5)-6

= (5/2)6



10. Find the value of n, when 3-7 ×32n + 3 = 311 ÷ 35

Solution:


32n + 3 = 311 ÷ 35/3-7

⇒ 32n + 3 = 311 - 5/3-7

⇒ 32n + 3 = 36/3-7

⇒ 32n + 3 = 36 - (-7)

⇒ 32n + 3 = 36 + 7

⇒ 32n + 3 = 313

Since the bases are same and equating the powers, we get 2n + 3 = 13

2n = 13 – 3

2n = 10

n = 10/2

Therefore, n = 5



11. Find the value of n, when (5/3)2n + 1 (5/3)5 = (5/3)n + 2

Solution:


(5/3)2n + 1 + 5 = (5/3)n + 2

= (5/3)2n + 6 = (5/3)n + 2

Since the bases are same and equating the powers, we get 2n + 6 = n + 2

2n – n = 2 – 6

=> n = -4



12. Find the value of n, when 3n = 243

Solution:


3n = 35

Since, the bases are same, so omitting the bases, and equating the powers we get, n = 5.

13. Find the value of n, when 271/n = 3

Solution:


(27) = 3n

⇒ (3)3 = 3n

Since, the bases are same and equating the powers, we get

⇒ n = 3



14. Find the value of n, when 3432/n = 49

Solution:


[(7)3]2/n = (7)2

⇒ (7)6/n = (7)2

⇒ 6/n = 2

Since, the bases are same and equating the powers, we get n = 6/2 = 3.



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