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                               
  ((————— ÷ s41598 -  20) -  67289) -  8
    5000                                

Step  2  :

         50519      
 Divide  —————  by  s41598
         5000       

Equation at the end of step  2  :

     (72•1031)                     
  ((—————————— -  20) -  67289) -  8
    5000s41598                      

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  5000s⁴¹⁵⁹⁸  as the denominator :

          20     20 • 5000s41598
    20 =  ——  =  ———————————————
          1        5000s41598   

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) - (20 • 5000s41598)     72•1031 - 100000s41598
 —————————————————————————————  =  ——————————————————————
          5000s41598                     5000s41598      

Equation at the end of step  3  :

   (72•1031 - 100000s41598)              
  (———————————————————————— -  67289) -  8
          5000s41598                     

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  5000s⁴¹⁵⁹⁸  as the denominator :

             67289     67289 • 5000s41598
    67289 =  —————  =  ——————————————————
               1           5000s41598    

Trying to factor as a Difference of Squares :

 4.2      Factoring:  50519 - 100000s⁴¹⁵⁹⁸ 

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 - 100000s⁴¹⁵⁹⁸ 

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

 (50519-100000s41598) - (67289 • 5000s41598)      50519 - 336545000s41598
 ———————————————————————————————————————————  =  ———————————————————————
                 5000s41598                            5000s41598       

Equation at the end of step  4  :

  (50519 - 336545000s41598)    
  ————————————————————————— -  8
         5000s41598            

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  5000s⁴¹⁵⁹⁸  as the denominator :

         8     8 • 5000s41598
    8 =  —  =  ——————————————
         1       5000s41598  

Trying to factor as a Sum of Cubes :

 5.2      Factoring:  50519 - 336545000s⁴¹⁵⁹⁸ 

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

Adding fractions that have a common denominator :

 5.3       Adding up the two equivalent fractions

 (50519-336545000s41598) - (8 • 5000s41598)      50519 - 336585000s41598
 ——————————————————————————————————————————  =  ———————————————————————
                 5000s41598                           5000s41598       

Trying to factor as a Difference of Squares :

 5.4      Factoring:  50519 - 336585000s⁴¹⁵⁹⁸ 

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:

 5.5      Factoring:  50519 - 336585000s⁴¹⁵⁹⁸ 

Check :  50519  is not a cube !!

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

Final result :

  50519 + 336585000s41598
  ———————————————————————
        5000s41598