Solution - Reducing fractions to their lowest terms

Step  1  :

            x27
 Simplify   ———
             h 

Equation at the end of step  1  :

                          x27
  (g•(((x23)•(x4))•h))-(g•———)
                           h 

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

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

              gx27h     gx27h • h
     gx27h =  —————  =  —————————
                1           h    

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:

 gx27h • h - (gx27)      gx27h2 - gx27 
 ——————————————————  =  —————————————
         h                    h      

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   gx²⁷h² - gx²⁷  =   gx²⁷ • (h² - 1) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  h² - 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 :  h²  is the square of  h¹ 

Factorization is :       (h + 1)  •  (h - 1) 

Final result :

  gx27 • (h + 1) • (h - 1)
  ————————————————————————
             h