Solution - Adding, subtracting and finding the least common multiple

Step  1  :

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

Trying to factor as a Difference of Squares :

 1.1      Factoring:  n² - m² 

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 :  n²  is the square of  n¹ 

Check :  m²  is the square of  m¹ 

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

Equation at the end of step  1  :

     1     1            m     
  (—————-—————)+(2•———————————)
   (m+n) (m-n)     (m+n)•(n-m)

Step  2  :

Equation at the end of step  2  :

     1     1         2m    
  (—————-—————)+———————————
   (m+n) (m-n)  (m+n)•(n-m)

Step  3  :

              1  
 Simplify   —————
            m - n

Equation at the end of step  3  :

     1    1        2m    
  (—————-———)+———————————
   (m+n) m-n  (m+n)•(n-m)

Step  4  :

              1  
 Simplify   —————
            m + n

Equation at the end of step  4  :

     1        1               2m       
  (————— -  —————) +  —————————————————
   m + n    m - n     (m + n) • (n - m)

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       m+n 

      The right denominator is :       m-n 

      Least Common Multiple:
      (m+n) • (m-n) 

Calculating Multipliers :

 5.2    Calculate multipliers for the two fractions


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = m-n

   Right_M = L.C.M / R_Deno = m+n

Making Equivalent Fractions :

 5.3      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent,  y/(y+1)²   and  (y²+y)/(y+1)³  are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

   L. Mult. • L. Num.           m-n     
   ——————————————————  =   —————————————
         L.C.M             (m+n) • (m-n)

   R. Mult. • R. Num.           m+n     
   ——————————————————  =   —————————————
         L.C.M             (m+n) • (m-n)

Adding fractions that have a common denominator :

 5.4       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:

  m-n - (m+n)             -2n       
 —————————————  =  —————————————————
 (m+n) • (m-n)     (m + n) • (m - n)

Equation at the end of step  5  :

         -2n                   2m       
  ————————————————— +  —————————————————
  (m + n) • (m - n)    (m + n) • (n - m)

Step  6  :

Making Equivalent Fractions :

 6.1      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.           -2n     
   ——————————————————  =   —————————————
         L.C.M             (m+n) • (m-n)

   R. Mult. • R. Num.         2m • -1   
   ——————————————————  =   —————————————
         L.C.M             (m+n) • (m-n)

Adding fractions that have a common denominator :

 6.2       Adding up the two equivalent fractions

 -2n + 2m • -1          -2m - 2n    
 —————————————  =  —————————————————
 (m+n) • (m-n)     (m + n) • (m - n)

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   -2m - 2n  =   -2 • (m + n) 

Canceling Out :

 7.2    Cancel out  (m + n)  which appears on both sides of the fraction line.

Final result :

    +2 
  —————
  m + n