Solution - Reducing fractions to their lowest terms

Step  1  :

             m
 Simplify   ——
            m2

Dividing exponential expressions :

 1.1    m¹ divided by m² = m^{(1 - 2)} = m^(-1) = ¹/_m¹ = ¹/_m

Equation at the end of step  1  :

   ((m2)-(n2))   1
  (———————————-n)—-mn)
        m     (  m

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a whole from a fraction

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

          mn     mn • m
    mn =  ——  =  ——————
          1        m   

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:

 1 - (mn • m)     1 - m2n
 ————————————  =  ———————
      m              m   

Equation at the end of step  2  :

   ((m2)-(n2))   (1-m2n)
  (———————————-n)———————
        m           m   

Step  3  :

            m2 - n2
 Simplify   ———————
               m   

Trying to factor as a Difference of Squares :

 3.1      Factoring:  m² - n² 

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 :  m²  is the square of  m¹ 

Check :  n²  is the square of  n¹ 

Factorization is :       (m + n)  •  (m - n) 

Equation at the end of step  3  :

  (m + n) • (m - n)        (1 - m2n)
  ————————————————— - n) ÷ —————————
          m                    m    

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

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

         n     n • m
    n =  —  =  —————
         1       m  

Adding fractions that have a common denominator :

 4.2       Adding up the two equivalent fractions

 (m+n) • (m-n) - (n • m)     m2 - mn - n2
 ———————————————————————  =  ————————————
            m                     m      

Equation at the end of step  4  :

  (m2 - mn - n2)   (1 - m2n)
  —————————————— ÷ —————————
        m              m    

Step  5  :

         m2-mn-n2      1-m2n
 Divide  ————————  by  —————
            m            m  

 5.1    Dividing fractions

To divide fractions, write the divison as multiplication by the reciprocal of the divisor :

m2 - mn - n2     1 - m2n       m2 - mn - n2         m    
————————————  ÷  ———————   =   ————————————  •  —————————
     m              m               m           (1 - m2n)

Trying to factor a multi variable polynomial :

 5.2    Factoring    m² - mn - n² 

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

 Factorization fails

Trying to factor as a Difference of Squares :

 5.3      Factoring:  1 - m²n 

Check :  1  is the square of  1 
Check :  m²  is the square of  m¹ 

Check :  n¹   is not a square !!
Ruling : Binomial can not be factored as the difference of two perfect squares

Final result :

  m2 + mn + n2
  ————————————
    1 + m2n