Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "^-5" was replaced by "^(-5)". 1 more similar replacement(s)

(2): "6.4" was replaced by "(64/10)".

Step  1  :

 1.1     10 = 2•5

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

Equation at the end of step  1  :

                   64
  (7 • (10-1)) +  (—— • ((2)(-5)•(5)(-5)))
                   10

Step  2  :

            32
 Simplify   ——
            5 

Equation at the end of step  2  :

                   32
  (7 • (10-1)) +  (—— • ((2)(-5)•(5)(-5)))
                   5 

Step  3  :

Multiplying exponents :

 3.1    5¹  multiplied by  5⁵   = 5^{(1 + 5)} = 5⁶

Raising to a Power :

 3.2      Canceling out  2⁵ as it appears on both sides of the fraction line

Equation at the end of step  3  :

                   1
  (7 • (10-1)) +  ——
                  56

Step  4  :

 4.1     10 = 2•5

(10)^-1 = (2•5)^(-1) = (2)^(-1) • (5)^(-1)

Equation at the end of step  4  :

                              1
  (7 • ((2)(-1)•(5)(-1))) +  ——
                             56

Step  5  :

Equation at the end of step  5  :

   7     1
  —— +  ——
  10    56

Step  6  :

 6.1      Finding a Common Denominator   The left  10 
 The right  56 

 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. 

 10 • 56   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 = 5⁶

   Right_M = L.C.M / R_Deno = 10

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.                   7 • 56
   ——————————————————  =               ———————
             Common denominator        10 • 56

   R. Mult. • R. Num.                     10  
   ——————————————————  =               ———————
             Common denominator        10 • 56

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:

 7 • 56 + 10     7•56 + 10
 ———————————  =  —————————
   10 • 56        10 • 56 

Final result :

   7 + 10
  ———————
  10 • 56