Solution - Reducing fractions to their lowest terms

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "10.1038" was replaced by "(101038/10000)".

Step  1  :

            50519
 Simplify   —————
            5000 

Equation at the end of step  1  :

    50519                             
  ((————— ÷ d41586 -  25) -  405) -  8
    5000                              

Step  2  :

         50519      
 Divide  —————  by  d41586
         5000       

Equation at the end of step  2  :

     (72•1031)                   
  ((—————————— -  25) -  405) -  8
    5000d41586                    

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  5000d⁴¹⁵⁸⁶  as the denominator :

          25     25 • 5000d41586
    25 =  ——  =  ———————————————
          1        5000d41586   

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 3.2       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:

 (72•1031) - (25 • 5000d41586)     72•1031 - 125000d41586
 —————————————————————————————  =  ——————————————————————
          5000d41586                     5000d41586      

Equation at the end of step  3  :

   (72•1031 - 125000d41586)            
  (———————————————————————— -  405) -  8
          5000d41586                   

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  5000d⁴¹⁵⁸⁶  as the denominator :

           405     405 • 5000d41586
    405 =  ———  =  ————————————————
            1         5000d41586   

Trying to factor as a Difference of Squares :

 4.2      Factoring:  50519 - 125000d⁴¹⁵⁸⁶ 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check :  50519  is not a square !!

Ruling : Binomial can not be factored as the
difference of two perfect squares

Trying to factor as a Difference of Cubes:

 4.3      Factoring:  50519 - 125000d⁴¹⁵⁸⁶ 

Theory : A difference of two perfect cubes,  a³ - b³ can be factored into
              (a-b) • (a² +ab +b²)

Proof :  (a-b)•(a²+ab+b²) =
            a³+a²b+ab²-ba²-b²a-b³ =
            a³+(a²b-ba²)+(ab²-b²a)-b³ =
            a³+0+0-b³ =
            a³-b³

Check :  50519  is not a cube !!

Ruling : Binomial can not be factored as the difference of two perfect cubes

Adding fractions that have a common denominator :

 4.4       Adding up the two equivalent fractions

 (50519-125000d41586) - (405 • 5000d41586)      50519 - 2150000d41586
 —————————————————————————————————————————  =  —————————————————————
                5000d41586                          5000d41586      

Equation at the end of step  4  :

  (50519 - 2150000d41586)    
  ——————————————————————— -  8
        5000d41586           

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  5000d⁴¹⁵⁸⁶  as the denominator :

         8     8 • 5000d41586
    8 =  —  =  ——————————————
         1       5000d41586  

Trying to factor as a Difference of Squares :

 5.2      Factoring:  50519 - 2150000d⁴¹⁵⁸⁶ 

Check :  50519  is not a square !!

Ruling : Binomial can not be factored as the
difference of two perfect squares

Trying to factor as a Difference of Cubes:

 5.3      Factoring:  50519 - 2150000d⁴¹⁵⁸⁶ 

Check :  50519  is not a cube !!

Ruling : Binomial can not be factored as the difference of two perfect cubes

Adding fractions that have a common denominator :

 5.4       Adding up the two equivalent fractions

 (50519-2150000d41586) - (8 • 5000d41586)      50519 - 2190000d41586
 ————————————————————————————————————————  =  —————————————————————
                5000d41586                         5000d41586      

Trying to factor as a Sum of Cubes :

 5.5      Factoring:  50519 - 2190000d⁴¹⁵⁸⁶ 

Theory : A sum of two perfect cubes,  a³ + b³ can be factored into  :
             (a+b) • (a²-ab+b²)
Proof  : (a+b) • (a²-ab+b²) =
    a³-a²b+ab²+ba²-b²a+b³ =
    a³+(a²b-ba²)+(ab²-b²a)+b³=
    a³+0+0+b³=
    a³+b³

Check :  50519  is not a cube !!

Ruling : Binomial can not be factored as the difference of two perfect cubes

Final result :

  50519 - 2190000d41586
  —————————————————————
       5000d41586