Class 8 NCERT Solutions- Chapter 12 Exponents and Powers – Exercise 12.1

Question 1. Evaluate:

Solution:

(i) 3^–2

3^-2 =  (Property used: a^-n = )

(ii) (– 4)^{– 2}

(-4)^-2 =  (Property used: a^-n = )

(iii) () ^-5 

()^-5  = (2)⁵ = 32 (Property used: )

Question 2. Simplify and express the result in power notation with a positive exponent.

Solution:

(i) (-4)⁵ ÷ (-4)⁸

= (-4)^5-8 = (-4) ^-3                     (Property used: a^m ÷ aⁿ= a^m-n)

= 

= 

(ii) 

=  (Property used: (a^m)ⁿ = a^m×n)

= 

= 

(iii) (-3)⁴ × ()⁴

= ((3)⁴ ×  (Property used: (a/b)ⁿ = aⁿ/ bⁿ  & (-a)ⁿ = aⁿ if a is positive number and n is even)

= 5⁴

(iv) (3^-7 ÷ 3^-10) × 3^-5

= 3 ^(-7-(-10)) × 3^-5                                    (Property used: a^m ÷ aⁿ= a^m-n)

= 3 ^(-7+10) × 3^-5

= 3³ × 3^-5

= 3 ^(3+(-5))                                                  (Property used: a^m × aⁿ = a ^{m + n})

= 3^-2                                                        (Property used: a^-m  =)

= 

= 

(v) 2^-3 × (-7)^-3

= (2 × (-7))^-3                                          (Property used: a^m × b^m = (a×b)^m)

= (-14)^-3                                                    (Property used: a^-m  =)

= 

=

Question 3. Find the value of

Solution:

(i) (3⁰ + 4^-1) × 2²

= (1 + ()) × 4                     (a⁰ = 1  a ≠ 0)

= () × 4

= 5

(ii) (2^-1 × 4^-1) ÷ 2^-2

= (2 × 4)^-1 ÷  (Property used: a^m × b^m = (a×b)^m)

= (8)-1 ÷ 

= () ÷ 

=() × 4

= ()

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

= 2² + 3² + 4²                                       (Property used:   =a^m)

= 4 + 9 + 16

= 29

(iv) (3^-1 + 4^-1 + 5^-1)⁰

= ()⁰                            (a⁰ = 1 (a ≠ 0)

= 0 

(v) {()^-2}²

= () ^-2×2                                    (Property used: (a^m)ⁿ = a^m×n) 

= ()^-4    = ()⁴                     (Property used: (b/a)^-n = aⁿ/bⁿ)

= 

= 

Question 4. Evaluate 

Solution:

(i) (8^-1 × 5³) / 2^-4

= ( × 125) / (2^-4)                      (Property used: (b/a)^-n = aⁿ/bⁿ)

= () × 125 × 2⁴

= 250

(ii) (5^-1 × 2^-1) × 6^-1

= (5 × 2)^-1 × 6^-1                          (Property used: a^m × b^m = (a×b)^m)

= 10^-1 × 6^-1

= (10 × 6)^-1                                     (Property used: a^m × b^m = (a×b)^m)

= 60^-1

= 

Question 5. Find the value of m for which 5^m ÷ 5^{– 3} = 5⁵ 

Solution:

5^{m-(– 3)} = 5⁵                       (Property used: a^m ÷ aⁿ= a^m-n)

5^m+3 = 5⁵ 

m+3 = 5

m = 5-3

m = 2

Question 6. Evaluate 

Solution:

(i) {()^-1 – ()^-1}^-1

= (3¹ – 4¹) ^-1                         (Property used: (1/a)^-m  = a^m)

= (-1)^-1

= (1/(-1))¹

= (-1)

(ii) ()^-7 × ()^-4

= ()⁷ × ()^-4                       (Property used: (b/a)^-n = (a/b)ⁿ)

= () ^{7+ (-4)}                               (Property used: a^m × aⁿ = a ^{m + n})

= ()³  = 8³/5³

= 

Question 7. Simplify

Solution:

(i)  (t ≠ 0)

=  (Property used: a^m ÷ aⁿ= a^m-n ) (25 = 5²)

=

= 

= 

(ii) 

= 

=  (Property used: (a×b)^m = a^m × b^m)

= (3^-5-(-5) × 2^-5-(-5) × 5 ^(-5)+3+7) (Property used: a^m ÷ aⁿ= a^m-n )

= (3⁰ × 2⁰ × 5⁵) (a⁰ = 1 (a ≠ 0)

= 5⁵