Fractional Exponents:

Fractional exponents are defined as follows, a^1/m ≡ ⁿ√a. This permits manipulations with numbers with fractional exponents to be treated using the laws expressed earlier for integers.  For instance,

8^1/3 ≡ ³√8 = 2   since 2×2×2 = 8

Taking  the  statement 8^1/3 =2 and  cubing  both  sides, (8^1/3)³  =2³  But  (a^m)ⁿ   =  a^{m  x  n}      so (8^1/3)³  =8¹ = 8 which agrees with 2³  = 8 for the right-hand side of the equality^.

A number such as 8^2/3 can be written (8^1/3)² =2² = 4 or alternately as (8²)^1/3 = (64)^1/3 =4 since 4×4×4 =64; that is 4 is the cube root of 64.

Examples:

(a^1/3) . (a^2/3) = a^{(1/3 . 2/3)} = a¹ =a

B^1/4/b^1/2 = b^(1/4-1/2) = b^-1/2 = 1/b²

(d^1/3)⁹ = d^(1/3×9) = d³