Number Systems (Exercise 1.6)

Number Systems

Chapter 1

Exercise 1.6

EX 1.6 QUESTION 1.

Find:

(i)64^1/2

(ii)32^1/5

(iii)125^1/3

Solution:

(i)64^1/2

64^1/2

= (8×8)^1/2

= (8²)^½

= 8¹ [⸪2×1/2 = 2/2 =1]

= 8

(ii)32^1/5

32^1/5 

= (2⁵)^1/5

= (2⁵)^⅕

= 2¹ [⸪5×1/5 = 1]

= 2

(iii)125^1/3

(125)^1/3

= (5×5×5)^1/3

= (5³)^⅓

= 5¹ (3×1/3 = 3/3 = 1)

= 5

EX 1.6 QUESTION 2.

 Find:

(i) 9^3/2

(ii) 32^2/5

(iii)16^3/4

(iv) 125^-1/3

Solution:

(i) 9^3/2

9^3/2

= (3×3)^3/2

= (3²)^3/2

= 3³ [⸪2×3/2 = 3]

=27

(ii) 32^2/5

32^2/5 = (2×2×2×2×2)^2/5

= (2⁵)^2⁄5

= 2² [⸪5×2/5= 2]

= 4

(iii)16^3/4

16^3/4 

= (2×2×2×2)^3/4

= (2⁴)^3⁄4

= 2³ [⸪4×3/4 = 3]

= 8

(iv) 125^-1/3

125^-1/3 

= (5×5×5)^-1/3

= (5³)^-1⁄3

= 5^-1 [⸪3×-1/3 = -1]

= 1/5

EX 1.6 QUESTION 3.

Simplify:

(i) 2^2/3×2^1/5

(ii) (1/3³)⁷

(iii) 11^1/2/11^1/4

(iv) 7^1/2×8^1/

Solution:

(i) 2^2/3×2^1/5

2^2/3×2^1/5 

= 2^(2/3)+(1/5)                   [ a^m×aⁿ=a^m+n]

= 2^13/15 [⸪2/3 + 1/5 = (2×5+3×1)/(3×5) = 13/15]

(ii) (1/3³)⁷

(1/3³)⁷ = (3^-3)⁷                [(a^m)ⁿ = a^{m x n}]

= 3^-21

(iii) 11^1/2/11^1/4

11^1/2/11^1/4 = 11^(1/2)-(1/4)

= 11^1/4 [⸪(1/2) – (1/4)

= (1×4-2×1)/(2×4) = 4-2)/8 = 2/8 = ¼ ]

(iv) 7^1/2×8^1/2

7^1/2×8^1/2 = (7×8)^1/2             [a^m×b^m = (a×b)^m ]

= 56^1/2