Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "b2"   was replaced by   "b^2".  1 more similar replacement(s).

Step  1  :

            a
 Simplify   —
            3

Equation at the end of step  1  :

        b               a
  ((((a•—)•(a2))-(b2))+(—•a))+b
        9               3
 Step  2  :
            b
 Simplify   —
            9

Equation at the end of step  2  :

          b                  a2     
  ((((a • —) • a2) -  b2) +  ——) +  b
          9                  3      

Step  3  :

Multiplying exponential expressions :

 3.1    a¹ multiplied by a² = a^{(1 + 2)} = a³

Equation at the end of step  3  :

    a3b           a2     
  ((——— -  b2) +  ——) +  b
     9            3      

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  9  as the denominator :

          b2     b2 • 9
    b2 =  ——  =  ——————
          1        9   

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:

 a3b - (b2 • 9)     a3b - 9b2
 ——————————————  =  —————————
       9                9    

Equation at the end of step  4  :

   (a3b - 9b2)    a2     
  (——————————— +  ——) +  b
        9         3      

Step  5  :

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   a³b - 9b²  =   b • (a³ - 9b) 

Trying to factor as a Difference of Cubes:

 6.2      Factoring:  a³ - 9b 

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 :  9  is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Calculating the Least Common Multiple :

 6.3    Find the Least Common Multiple

      The left denominator is :       9 

      The right denominator is :       3 

      Least Common Multiple:
      9 

Calculating Multipliers :

 6.4    Calculate multipliers for the two fractions


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 1

   Right_M = L.C.M / R_Deno = 3

Making Equivalent Fractions :

 6.5      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent,  y/(y+1)²   and  (y²+y)/(y+1)³  are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

   L. Mult. • L. Num.      b • (a3-9b)
   ——————————————————  =   ———————————
         L.C.M                  9     

   R. Mult. • R. Num.      a2 • 3
   ——————————————————  =   ——————
         L.C.M               9   

Adding fractions that have a common denominator :

 6.6       Adding up the two equivalent fractions

 b • (a3-9b) + a2 • 3     a3b + 3a2 - 9b2 
 ————————————————————  =  ———————————————
          9                      9       

Equation at the end of step  6  :

  (a3b + 3a2 - 9b2)     
  ————————————————— +  b
          9            

Step  7  :

Rewriting the whole as an Equivalent Fraction :

 7.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  9  as the denominator :

         b     b • 9
    b =  —  =  —————
         1       9  

Trying to factor a multi variable polynomial :

 7.2    Factoring    a³b + 3a² - 9b² 

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

 Factorization fails

Adding fractions that have a common denominator :

 7.3       Adding up the two equivalent fractions

 (a3b+3a2-9b2) + b • 9      a3b + 3a2 - 9b2 + 9b 
 —————————————————————  =  ————————————————————
           9                        9          

Checking for a perfect cube :

 7.4    a³b + 3a² + 9b² + 9b  is not a perfect cube

Final result :

 a3b + 3a2 + 9b2 + 9b