Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "^-4" was replaced by "^(-4)".

(2): "1.24" was replaced by "(124/100)". 2 more similar replacement(s)

Step  1  :

 1.1     10 = 2•5

(10)^-4 = (2•5)^(-4) = (2)^(-4) • (5)^(-4)

Equation at the end of step  1  :

     35        124
  ———————— +  (——— • ((2)(-4)•(5)(-4)))
  10000000     100

Step  2  :

            31
 Simplify   ——
            25

Equation at the end of step  2  :

     35        31
  ———————— +  (—— • ((2)(-4)•(5)(-4)))
  10000000     25

Step  3  :

Multiplying exponents :

 3.1    5²  multiplied by  5⁴   = 5^{(2 + 4)} = 5⁶

Equation at the end of step  3  :

     35          31  
  ———————— +  ———————
  10000000    (56•24)

Step  4  :

               7   
 Simplify   ———————
            2000000

Equation at the end of step  4  :

     7          31  
  ——————— +  ———————
  2000000    (56•24)

Step  5  :

 5.1      Finding a Common Denominator   The left  2000000 
 The right  56 • 24 

 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. 

 2000000 • 56 • 24   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 = 5⁶ • 2⁴

   Right_M = L.C.M / R_Deno = 2000000

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.                     7 • (56•24)   
   ——————————————————  =               —————————————————
             Common denominator        2000000 • (56•24)

   R. Mult. • R. Num.                     31 • 2000000  
   ——————————————————  =               —————————————————
             Common denominator        2000000 • (56•24)

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:

 7 • (56•24) + 31 • 2000000     7•56•24 + 62000000
 ——————————————————————————  =  ——————————————————
     2000000 • (56•24)          2000000 • (56•24) 

Final result :

     7 + 62000000  
  —————————————————
  2000000 • (56•24)