Solution - Reducing fractions to their lowest terms

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "a1"   was replaced by   "a^1". 

Step  1  :

             q
 Simplify   ——
            16

Equation at the end of step  1  :

     q                   
  ((—— • a1972) • s) -  2014
    16                   

Step  2  :

Equation at the end of step  2  :

   qa1972          
  (—————— • s) -  2014
     16           

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  16  as the denominator :

            2014     2014 • 16
    2014 =  ————  =  —————————
             1          16    

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:

 qa1972s - (2014 • 16)      qa1972s - 32224 
 —————————————————————  =  ———————————————
          16                     16       

Trying to factor as a Difference of Squares :

 3.3      Factoring:  qa¹⁹⁷²s - 32224 

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 : 32224 is not a square !!

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

Final result :

  qa1972s - 32224 
  ———————————————
        16