Solution - Reducing fractions to their lowest terms

Step  1  :

            b5
 Simplify   ——
            a2

Equation at the end of step  1  :

                                                                   b5
  (((((((a6)+(b6))-((a5)•b))-((4•(a4))•(b2)))+((6•(a3))•(b3)))-(3a•——))-2ab)+b2
                                                                   a2
 Step  2  :

Equation at the end of step  2  :

                                                            3b5
  (((((((a6)+(b6))-((a5)•b))-((4•(a4))•(b2)))+((2•3a3)•b3))-———)-2ab)+b2
                                                             a 
 Step  3  :

Equation at the end of step  3  :

                                                   3b5
  (((((((a6)+(b6))-((a5)•b))-(22a4•b2))+(2•3a3b3))-———)-2ab)+b2
                                                    a 

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a fraction from a whole

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

                                      a6 - a5b - 4a4b2 + 6a3b3 + b6      (a6 - a5b - 4a4b2 + 6a3b3 + b6) • a 
     a6 - a5b - 4a4b2 + 6a3b3 + b6 =  —————————————————————————————  =  ———————————————————————————————————
                                                    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 :

 4.2       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:

 (a6-a5b-4a4b2+6a3b3+b6) • a - (3b5)      a7 - a6b - 4a5b2 + 6a4b3 + ab6 - 3b5 
 ———————————————————————————————————  =  ————————————————————————————————————
                  a                                       a                  

Equation at the end of step  4  :

   (a7 - a6b - 4a5b2 + 6a4b3 + ab6 - 3b5)             
  (—————————————————————————————————————— -  2ab) +  b2
                     a                               

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction

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

           2ab     2ab • a
    2ab =  ———  =  ———————
            1         a   

Trying to factor by pulling out :

 5.2      Factoring:  a⁷ - a⁶b - 4a⁵b² + 6a⁴b³ + ab⁶ - 3b⁵ 

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  -a⁶b + a⁷ 
Group 2:  -4a⁵b² + 6a⁴b³ 
Group 3:  ab⁶ - 3b⁵ 

Pull out from each group separately :

Group 1:   (a - b) • (a⁶)
Group 2:   (2a - 3b) • (-2a⁴b²)
Group 3:   (ab - 3) • (b⁵)


Looking for common sub-expressions :

Group 1:   (a - b) • (a⁶)
Group 3:   (ab - 3) • (b⁵)
Group 2:   (2a - 3b) • (-2a⁴b²)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Adding fractions that have a common denominator :

 5.3       Adding up the two equivalent fractions

 (a7-a6b-4a5b2+6a4b3+ab6-3b5) - (2ab • a)      a7 - a6b - 4a5b2 + 6a4b3 - 2a2b + ab6 - 3b5 
 ————————————————————————————————————————  =  ———————————————————————————————————————————
                    a                                              a                     

Equation at the end of step  5  :

  (a7 - a6b - 4a5b2 + 6a4b3 - 2a2b + ab6 - 3b5)     
  ————————————————————————————————————————————— +  b2
                        a                          

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Adding a whole to a fraction

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

          b2     b2 • a
    b2 =  ——  =  ——————
          1        a   

Adding fractions that have a common denominator :

 6.2       Adding up the two equivalent fractions

 (a7-a6b-4a5b2+6a4b3-2a2b+ab6-3b5) + b2 • a      a7 - a6b - 4a5b2 + 6a4b3 - 2a2b + ab6 + ab2 - 3b5 
 ——————————————————————————————————————————  =  —————————————————————————————————————————————————
                     a                                                  a                        

Final result :

  a7 - a6b - 4a5b2 + 6a4b3 - 2a2b + ab6 + ab2 - 3b5 
  —————————————————————————————————————————————————
                          a