Solution - Reducing fractions to their lowest terms

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "b5"   was replaced by   "b^5".  1 more similar replacement(s).

Step  1  :

Equation at the end of step  1  :

             (b4)      
  (a -  (3 • ————)) -  5b5
              a        
 Step  2  :
            b4
 Simplify   ——
            a 

Equation at the end of step  2  :

             b4      
  (a -  (3 • ——)) -  5b5
             a       

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

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

          a     a • a
     a =  —  =  —————
          1       a  

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:

 a • a - (3b4)     a2 - 3b4
 —————————————  =  ————————
       a              a    

Equation at the end of step  3  :

  (a2 - 3b4)    
  —————————— -  5b5
      a         

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

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

           5b5     5b5 • a
    5b5 =  ———  =  ———————
            1         a   

Trying to factor as a Difference of Squares :

 4.2      Factoring:  a² - 3b⁴ 

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 : 3 is not a square !!

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

Adding fractions that have a common denominator :

 4.3       Adding up the two equivalent fractions

 (a2-3b4) - (5b5 • a)      a2 - 5ab5 - 3b4 
 ————————————————————  =  ———————————————
          a                      a       

Trying to factor a multi variable polynomial :

 4.4    Factoring    a² - 5ab⁵ - 3b⁴ 

Try to factor this multi-variable trinomial using trial and error 

 Factorization fails

Final result :

  a2 + 5ab5 + 3b4 
  ———————————————
         a