1. Find:

(i)64 ^1/2

Solution:

= 64 ^1/2 = (8×8) ^1/2

= (8²)^1/2

= 8¹ [2×1/2 = 2/2 =1]

= 8

(ii)32 ^1/5

Solution:

= 32 ^1/5 = (2⁵)^1/5

= (2⁵) ^1/5

= 2¹ [5×1/5 = 1]

= 2

(iii)125 ^1/3

Solution:

= (125) ^1/3 = (5×5×5)^1/3

= (5³)^1/3

= 5¹ (3×1/3 = 3/3 = 1)

= 5

2. Find:

(i) 9^3/2

Solution:

=9^3/2 = (3×3)3/2

= (3²)^3/2

= 3³ [2×3/2 = 3]

=27

(ii) 32^2/5

Solution:

=32^2/5 = (2×2×2×2×2)2/5

= (2⁵)^2⁄5

= 2² [5×2/5= 2]

= 4

(iii)16^3/4

Solution:

= 16^3/4 = (2×2×2×2)^3/4

= (2⁴)^3⁄4

= 2³ [4×3/4 = 3]

= 8

(iv) 125 ^(-1/3)

Solution:

= 125 ^-1/3 = (5×5×5) ^-1/3

= (5³) ^-1⁄3

= 5-1 [3×-1/3 = -1]

= 1/5

3. Simplify:

(i) 2 ^2/3 × 2 ^1/5

Solution:

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

[Here, 2/3 + 1/5 = (2×5+3×1)/(3×5) = 13/15]

= 2^13/15

(ii) (1/3³)⁷

Solution:

= (1/3³)⁷ = (3 ^-3)⁷

= 3 ^-21

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

Solution:

= 11 ^1/2/11 ^1/4

= 11 ^(1/2)-(1/4)

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

= 11 ^1/4

(iv) 7 ^½ × 8 ^1/2

Solution:

7 ^½ × 8 ^1/2

= (7×8) ^1/2

= 56 ^1/2