Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "^-5" was replaced by "^(-5)".

(2): "0.0039" was replaced by "(0039/10000)".

Step  1  :

 1.1     10 = 2•5

(10)^-5 = (2•5)^(-5) = (2)^(-5) • (5)^(-5)

Equation at the end of step  1  :

    39     
  ————— -  (2 • ((2)(-5)•(5)(-5)))
  10000    

Step  2  :

Dividing exponents :

 2.1    2¹   divided by   2⁵   = 2^{(1 - 5)} = 2^(-4) = ¹/_2⁴

Equation at the end of step  2  :

    39        1   
  ————— -  ———————
  10000    (24•55)

Step  3  :

              39 
 Simplify   —————
            10000

Equation at the end of step  3  :

    39        1   
  ————— -  ———————
  10000    (24•55)

Step  4  :

 4.1      Finding a Common Denominator   The left  10000 
 The right  24 • 55 

 The product of any two denominators can be used
 as a common denominator. 

 Said product is not necessarily the least common 
 denominator.

 As a matter of fact, whenever the two denominators
have a common factor, their product will be bigger
 than the least common denominator. 

Anyway, the product is a fine common denominator and
 can perfectly be used for 
 calculating multipliers, as well as for generating
 equivalent fractions. 

 10000 • 24 • 55   will be used as a common denominator.

Calculating Multipliers :

 4.2    Calculate multipliers for the two fractions


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 2⁴ • 5⁵

   Right_M = L.C.M / R_Deno = 10000

Making Equivalent Fractions :

 4.3      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent,  y/(y+1)²   and  (y²+y)/(y+1)³  are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

   L. Mult. • L. Num.                    39 • (24•55) 
   ——————————————————  =               ———————————————
             Common denominator        10000 • (24•55)

   R. Mult. • R. Num.                       10000     
   ——————————————————  =               ———————————————
             Common denominator        10000 • (24•55)

Adding fractions that have a common denominator :

 4.4       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 39 • (24•55) - (10000)     3•13•24•55 - 10000
 ——————————————————————  =  ——————————————————
    10000 • (24•55)          10000 • (24•55)  

Final result :

     39 - 10000  
  ———————————————
  10000 • (24•55)