Solution - Reducing fractions to their lowest terms

Step  1  :

            x60 - 6
 Simplify   ———————
               3   

Trying to factor as a Difference of Squares :

 1.1      Factoring:  x⁶⁰ - 6 

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 : 6 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:

 1.2      Factoring:  x⁶⁰ - 6 

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

Equation at the end of step  1  :

   (x60 - 6)           
  (————————— • x2) -  7
       3               

Step  2  :

Equation at the end of step  2  :

  x2 • (x60 - 6)     
  —————————————— -  7
        3           

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  3  as the denominator :

         7     7 • 3
    7 =  —  =  —————
         1       3  

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:

 x2 • (x60-6) - (7 • 3)      x62 - 6x2 - 21 
 ——————————————————————  =  ——————————————
           3                      3       

Final result :

  x62 - 6x2 - 21 
  ——————————————
        3