Solution - Reducing fractions to their lowest terms

Step  1  :

             1
 Simplify   ——
            e2

Equation at the end of step  1  :

                  1
  ((e2) • x) -  (—— • 1)
                 e2

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  e²  as the denominator :

            e2x     e2x • e2
     e2x =  ———  =  ————————
             1         e2   

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 :

 2.2       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:

 e2x • e2 - (1)     e4x - 1
 ——————————————  =  ———————
       e2             e2   

Trying to factor as a Difference of Squares :

 2.3      Factoring:  e⁴x - 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 : 1 is the square of 1
Check :  e⁴  is the square of  e² 

Check :  x¹   is not a square !!
Ruling : Binomial can not be factored as the difference of two perfect squares

Final result :

  e4x - 1
  ———————
    e2