Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "^-6" was replaced by "^(-6)". 1 more similar replacement(s)

(2): "3.3" was replaced by "(33/10)".

Step  1  :

 1.1     10 = 2•5

(10)^-6 = (2•5)^(-6) = (2)^(-6) • (5)^(-6)

Equation at the end of step  1  :

                   33
  (5 • (10-3)) +  (—— • ((2)(-6)•(5)(-6)))
                   10

Step  2  :

            33
 Simplify   ——
            10

Equation at the end of step  2  :

                   33
  (5 • (10-3)) +  (—— • ((2)(-6)•(5)(-6)))
                   10

Step  3  :

Multiplying exponents :

 3.1    2¹  multiplied by  2⁶   = 2^{(1 + 6)} = 2⁷

Raising to a Power :

 3.2    5¹  multiplied by  5⁶   = 5^{(1 + 6)} = 5⁷

Equation at the end of step  3  :

                     33  
  (5 • (10-3)) +  ———————
                  (27•57)

Step  4  :

 4.1     10 = 2•5

(10)^-3 = (2•5)^(-3) = (2)^(-3) • (5)^(-3)

Equation at the end of step  4  :

                                33  
  (5 • ((2)(-3)•(5)(-3))) +  ———————
                             (27•57)

Step  5  :

Dividing exponents :

 5.1    5¹   divided by   5³   = 5^{(1 - 3)} = 5^(-2) = ¹/_5²

Equation at the end of step  5  :

   1        33  
  ——— +  ———————
  200    (27•57)

Step  6  :

 6.1      Finding a Common Denominator   The left  200 
 The right  27 • 57 

 The product of any two denominators can be used
 as a common denominator. 

 Said product is not necessarily the least common 
 denominator.

 As a matter of fact, whenever the two denominators
have a common factor, their product will be bigger
 than the least common denominator. 

Anyway, the product is a fine common denominator and
 can perfectly be used for 
 calculating multipliers, as well as for generating
 equivalent fractions. 

 200 • 27 • 57   will be used as a common denominator.

Calculating Multipliers :

 6.2    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 = 2⁷ • 5⁷

   Right_M = L.C.M / R_Deno = 200

Making Equivalent Fractions :

 6.3      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.                     (27•57)   
   ——————————————————  =               —————————————
             Common denominator        200 • (27•57)

   R. Mult. • R. Num.                     33 • 200  
   ——————————————————  =               —————————————
             Common denominator        200 • (27•57)

Adding fractions that have a common denominator :

 6.4       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:

 (27•57) + 33 • 200      27•57 + 6600
 ——————————————————  =  —————————————
   200 • (27•57)        200 • (27•57)

Final result :

     1 + 6600  
  —————————————
  200 • (27•57)