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): "2.5" was replaced by "(25/10)". 2 more similar replacement(s)

Step  1  :

 1.1     10 = 2•5

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

Equation at the end of step  1  :

    45       25
  —————— -  (—— • ((2)(-5)•(5)(-5)))
  100000     10

Step  2  :

            5
 Simplify   —
            2

Equation at the end of step  2  :

    45       5
  —————— -  (— • ((2)(-5)•(5)(-5)))
  100000     2

Step  3  :

Multiplying exponents :

 3.1    2¹  multiplied by  2⁵   = 2^{(1 + 5)} = 2⁶

Dividing exponents :

 3.2    5¹   divided by   5⁵   = 5^{(1 - 5)} = 5^(-4) = ¹/_5⁴

Equation at the end of step  3  :

    45         1   
  —————— -  ———————
  100000    (26•54)

Step  4  :

              9  
 Simplify   —————
            20000

Equation at the end of step  4  :

    9         1   
  ————— -  ———————
  20000    (26•54)

Step  5  :

 5.1      Finding a Common Denominator   The left  20000 
 The right  26 • 54 

 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. 

 20000 • 26 • 54   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 = 2⁶ • 5⁴

   Right_M = L.C.M / R_Deno = 20000

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.                    9 • (26•54)  
   ——————————————————  =               ———————————————
             Common denominator        20000 • (26•54)

   R. Mult. • R. Num.                       20000     
   ——————————————————  =               ———————————————
             Common denominator        20000 • (26•54)

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:

 9 • (26•54) - (20000)     32•26•54 - 20000
 —————————————————————  =  ————————————————
    20000 • (26•54)        20000 • (26•54) 

Final result :

     9 - 20000   
  ———————————————
  20000 • (26•54)