How do you simplify this expression?

Question

32 ^  4/5  -  32 ^ -4/5

Answer 1

 Expend each term, and simplify by arithmetic:

 

 

32^4/5 - 32^{-4/5 =} ( 32^1/5)⁴ - [1 /[(32^1/5 )⁴ ] ]

                                   2⁴    -  1/ 2⁴ =

                                   16  -   1/ 16 = (16 . 16 -1)/ 16

                                      255/ 16 = 15 15/16

      Note that fractional exponent is same as root;   i.e. ⁿ√X ^{n =} X

          32^1/5 =⁵√32= 5√(2.2.2. 2.2) = ⁵√2⁵ =2

Answer 2

Assuming you mean:

 

32^4/5 - 32^-4/5

 

(which makes some sense, since 32 has a "nice" 5th root of 2)

 

this is equal to:

 

(32^1/5)⁴ - (32^1/5)^-4      (exponent multiplication)

 

= (⁵√32)⁴ - (⁵√32)^-4    (fractional exponents are the same as roots)

 

= (2)⁴ - (2)^-4               (the 5th root of 32 is 2)

 

= 16 - 1/(2⁴)                (2⁴ = 16, and negative exponents are the same as the reciprocal of the positive exponent

 

= 16 - 1/16

 

= 256/16 - 1/16             (Common denominator)

 

= 255/15                       (fraction subtraction, improper fraction)

 

= 15 15/16                    (or as a mixed number)