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)".

Step  1  :

 1.1     10 = 2•5

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

Equation at the end of step  1  :

  (103) +  ((2)(-5)•(5)(-5))

Step  2  :

 2.1     10 = 2•5

(10)³ = (2•5)³ = 2³ • 5³

Equation at the end of step  2  :

  (23•53) +  ((2)(-5)•(5)(-5))

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  2⁵ • 5⁵  as the denominator :

               (23•53)     (23•53) • (25•55)
    (23•53) =  ———————  =  —————————————————
                  1             (25•55)     

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

Calculating Multipliers :

 3.2    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 = 2⁵ • 5⁵

   Right_M = L.C.M / R_Deno = 1

Adding fractions that have a common denominator :

 3.3       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:

Multiplying exponents :

 2³  multiplied by  2⁵   = 2^{(3 + 5)} = 2⁸

Multiplying exponents :

 5³  multiplied by  5⁵   = 5^{(3 + 5)} = 5⁸ (2³•5³) • (2⁵•5⁵) + 1 2⁸•5⁸ + 1 ————————————————————— = ————————— (2⁵•5⁵) 1

Final result :

  1000 + 1
  ————————
     1