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)". 1 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  :

  (250 • (10-6)) -  (7 • ((2)(-5)•(5)(-5)))

Step  2  :

Equation at the end of step  2  :

                       7   
  (250 • (10-6)) -  ———————
                    (25•55)

Step  3  :

 3.1     10 = 2•5

(10)^-6 = (2•5)^(-6) = (2)^(-6) • (5)^(-6)

Equation at the end of step  3  :

                                  7   
  (250 • ((2)(-6)•(5)(-6))) -  ———————
                               (25•55)

Step  4  :

Dividing exponents :

 4.1    2¹   divided by   2⁶   = 2^{(1 - 6)} = 2^(-5) = ¹/_2⁵

Dividing exponents :

 4.2    5³   divided by   5⁶   = 5^{(3 - 6)} = 5^(-3) = ¹/_5³

Equation at the end of step  4  :

    1        7   
  ———— -  ———————
  4000    (25•55)

Step  5  :

 5.1      Finding a Common Denominator   The left  4000 
 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. 

 4000 • 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 = 4000

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.                      (25•55)   
   ——————————————————  =               ——————————————
             Common denominator        4000 • (25•55)

   R. Mult. • R. Num.                     7 • 4000   
   ——————————————————  =               ——————————————
             Common denominator        4000 • (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:

 (25•55) - (7 • 4000)      25•55 - 28000
 ————————————————————  =  ——————————————
    4000 • (25•55)        4000 • (25•55)

Final result :

     1 - 28000  
  ——————————————
  4000 • (25•55)