Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "^-3" was replaced by "^(-3)".

(2): "3.31" was replaced by "(331/100)". 2 more similar replacement(s)

Step  1  :

 1.1     10 = 2•5

(10)^-3 = (2•5)^(-3) = (2)^(-3) • (5)^(-3)

Equation at the end of step  1  :

   899      331
  ————— +  (——— • ((2)(-3)•(5)(-3)))
  10000     100

Step  2  :

            331
 Simplify   ———
            100

Equation at the end of step  2  :

   899      331
  ————— +  (——— • ((2)(-3)•(5)(-3)))
  10000     100

Step  3  :

Multiplying exponents :

 3.1    2²  multiplied by  2³   = 2^{(2 + 3)} = 2⁵

Multiplying exponents :

 3.2    5²  multiplied by  5³   = 5^{(2 + 3)} = 5⁵

Equation at the end of step  3  :

   899       331  
  ————— +  ———————
  10000    (25•55)

Step  4  :

             899 
 Simplify   —————
            10000

Equation at the end of step  4  :

   899       331  
  ————— +  ———————
  10000    (25•55)

Step  5  :

 5.1      Finding a Common Denominator   The left  10000 
 The right  25 • 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 • 25 • 55   will be used as a common denominator.

Calculating Multipliers :

 5.2    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 = 2⁵ • 5⁵

   Right_M = L.C.M / R_Deno = 10000

Making Equivalent Fractions :

 5.3      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.                   899 • (25•55) 
   ——————————————————  =               ———————————————
             Common denominator        10000 • (25•55)

   R. Mult. • R. Num.                    331 • 10000  
   ——————————————————  =               ———————————————
             Common denominator        10000 • (25•55)

Adding fractions that have a common denominator :

 5.4       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:

 899 • (25•55) + 331 • 10000     29•31•25•55 + 3310000
 ———————————————————————————  =  —————————————————————
       10000 • (25•55)              10000 • (25•55)   

Final result :

   899 + 3310000 
  ———————————————
  10000 • (25•55)