Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "^-3" was replaced by "^(-3)". 1 more similar replacement(s)

(2): "3.9" was replaced by "(39/10)". 2 more similar replacement(s)

Step  1  :

 1.1     10 = 2•5

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

Equation at the end of step  1  :

   18          39
  (——•(10-2))-(——•((2)(-3)•(5)(-3)))
   10          10

Step  2  :

            39
 Simplify   ——
            10

Equation at the end of step  2  :

   18               39
  (—— • (10-2)) -  (—— • ((2)(-3)•(5)(-3)))
   10               10

Step  3  :

Multiplying exponents :

 3.1    2¹  multiplied by  2³   = 2^{(1 + 3)} = 2⁴

Raising to a Power :

 3.2    5¹  multiplied by  5³   = 5^{(1 + 3)} = 5⁴

Equation at the end of step  3  :

   18                 39  
  (—— • (10-2)) -  ———————
   10              (24•54)

Step  4  :

 4.1     10 = 2•5

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

Equation at the end of step  4  :

   18                            39  
  (—— • ((2)(-2)•(5)(-2))) -  ———————
   10                         (24•54)

Step  5  :

            9
 Simplify   —
            5

Equation at the end of step  5  :

   9                            39  
  (— • ((2)(-2)•(5)(-2))) -  ———————
   5                         (24•54)

Step  6  :

Multiplying exponents :

 6.1    5¹  multiplied by  5²   = 5^{(1 + 2)} = 5³

Equation at the end of step  6  :

   9        39  
  ——— -  ———————
  500    (24•54)

Step  7  :

 7.1      Finding a Common Denominator   The left  500 
 The right  24 • 54 

 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. 

 500 • 24 • 54   will be used as a common denominator.

Calculating Multipliers :

 7.2    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 = 2⁴ • 5⁴

   Right_M = L.C.M / R_Deno = 500

Making Equivalent Fractions :

 7.3      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.                   9 • (24•54) 
   ——————————————————  =               —————————————
             Common denominator        500 • (24•54)

   R. Mult. • R. Num.                     39 • 500  
   ——————————————————  =               —————————————
             Common denominator        500 • (24•54)

Adding fractions that have a common denominator :

 7.4       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:

 9 • (24•54) - (39 • 500)     32•24•54 - 19500
 ————————————————————————  =  ————————————————
      500 • (24•54)            500 • (24•54)  

Step  8  :

Pulling out like terms :

 8.1     Pull out like factors :

   3² • 2⁴ • 5⁴ - 19500  =   3 • (3 - 6500) 

Final result :

  3 • (3 + 6500)
  ——————————————
  500 • (24•54)