Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "^-1" was replaced by "^(-1))". 1 more similar replacement(s)

Step  1  :

            1
 Simplify   —
            n

Equation at the end of step  1  :

   ((xa)-11)                 1
  (————————— • ((xb)-1)) -  —
       n                    n
 Step  2  :
            x(-11)a(-11) 
 Simplify   ————————————
                 n      

Equation at the end of step  2  :

      1                     1
  (——————— • x(-1)b(-1)) -  —
   x11a11n                   n
 Step  3  :

Calculating the Least Common Multiple :

 3.1    Find the Least Common Multiple

      The left denominator is :       x¹²a¹¹nb 

      The right denominator is :       n 

      Least Common Multiple:
      x¹²a¹¹nb 

Calculating Multipliers :

 3.2    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 = x¹²a¹¹b

Making Equivalent Fractions :

 3.3      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.          1   
   ——————————————————  =   ————————
         L.C.M             x12a11nb 

   R. Mult. • R. Num.       x12a11b
   ——————————————————  =   ————————
         L.C.M             x12a11nb

Adding fractions that have a common denominator :

 3.4       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:

 1 - (x12a11b)     1 - x12a11b
 —————————————  =  ———————————
   x12a11nb         x12a11nb  

Trying to factor as a Difference of Squares :

 3.5      Factoring:  1 - x¹²a¹¹b 

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 :  1  is the square of  1 
Check :  x¹²  is the square of  x⁶ 

Check :  a¹¹   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:

 3.6      Factoring:  1 - x¹²a¹¹b 

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 :  1  is the cube of  1 

Check :  x¹² is the cube of   x⁴

Check :  a ¹¹ is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Final result :

  1 - x12a11b
  ———————————
   x12a11nb