Solution - Reducing fractions to their lowest terms

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "l5"   was replaced by   "l^5". 

Step  1  :

            18
 Simplify   ——
            l 

Equation at the end of step  1  :

                      18
  ((l-((7lx•(l5))•l))+——)-6l
                      l 
 Step  2  :

Multiplying exponential expressions :

 2.1    l¹ multiplied by l⁵ = l^{(1 + 5)} = l⁶

Equation at the end of step  2  :

                        18     
  ((l -  (7l6x • l)) +  ——) -  6l
                        l      

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a fraction to a whole

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

                 l - 7l7x     (l - 7l7x) • l
     l - 7l7x =  ————————  =  ——————————————
                    1               l       

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

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   l - 7l⁷x  =   -l • (7l⁶x - 1) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  7l⁶x - 1 

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 :  7  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:

 4.3      Factoring:  7l⁶x - 1 

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 :  7  is not a cube !!

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

Adding fractions that have a common denominator :

 4.4       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:

 -l • (7l6x-1) • l + 18     -7l8x + l2 + 18
 ——————————————————————  =  ———————————————
           l                       l       

Equation at the end of step  4  :

  (-7l8x + l2 + 18)    
  ————————————————— -  6l
          l            

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction

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

          6l     6l • l
    6l =  ——  =  ——————
          1        l   

Trying to factor a multi variable polynomial :

 5.2    Factoring    -7l⁸x + l² + 18 

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

 Factorization fails

Adding fractions that have a common denominator :

 5.3       Adding up the two equivalent fractions

 (-7l8x+l2+18) - (6l • l)     -7l8x - 5l2 + 18
 ————————————————————————  =  ————————————————
            l                        l        

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   -7l⁸x - 5l² + 18  =   -1 • (7l⁸x + 5l² - 18) 

Trying to factor a multi variable polynomial :

 6.2    Factoring    7l⁸x + 5l² - 18 

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

 Factorization fails

Final result :

  +7l8x + 5l2 + 18
  ————————————————
         l