What is 3x 8y+4x+2y

 (6x+3y)/4x2y-(3x-6y)/xy2 

Final result :

3y • (2x4 + x3y - 4xy + 8y2) ———————————————————————————— 4x

Step by step solution :

Step  1  :

3x - 6y Simplify ——————— x

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   3x - 6y  =   3 • (x - 2y) 

Equation at the end of step  2  :

(6x+3y) 3•(x-2y) ((———————•(x2))•y)-(————————•y2) 4 x

Step  3  :

Equation at the end of step  3  :

(6x+3y) 3y2•(x-2y) ((———————•(x2))•y)-—————————— 4 x

Step  4  :

6x + 3y Simplify ——————— 4

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   6x + 3y  =   3 • (2x + y) 

Equation at the end of step  5  :

3•(2x+y) 3y2•(x-2y) ((————————•x2)•y)-—————————— 4 x

Step  6  :

Equation at the end of step  6  :

3x2 • (2x + y) 3y2 • (x - 2y) (—————————————— • y) - —————————————— 4 x

Step  7  :

Equation at the end of step  7  :

3x2y • (2x + y) 3y2 • (x - 2y) ——————————————— - —————————————— 4 x

Step  8  :

Calculating the Least Common Multiple :

 8.1    Find the Least Common Multiple

      The left denominator is :       4 

      The right denominator is :       x 

 Prime 
 Factor 
 Left 
 Denominator 
 Right 
 Denominator 
 L.C.M = Max 
 {Left,Right} 
2202
 Product of all 
 Prime Factors 
414
    Algebraic    
    Factor    
 Left 
 Denominator 
 Right 
 Denominator 
 L.C.M = Max 
 {Left,Right} 
 x 011


      Least Common Multiple:
      4x 

Calculating Multipliers :

 8.2    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 = x

   Right_M = L.C.M / R_Deno = 4

Making Equivalent Fractions :

 8.3      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. 3x2y • (2x+y) • x —————————————————— = ————————————————— L.C.M 4x R. Mult. • R. Num. 3y2 • (x-2y) • 4 —————————————————— = ———————————————— L.C.M 4x

Adding fractions that have a common denominator :

 8.4       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:

3x2y • (2x+y) • x - (3y2 • (x-2y) • 4) 6x4y + 3x3y2 - 12xy2 + 24y3 —————————————————————————————————————— = ——————————————————————————— 4x 4x

Step  9  :

Pulling out like terms :

 9.1     Pull out like factors :

   6x4y + 3x3y2 - 12xy2 + 24y3  = 

  3y • (2x4 + x3y - 4xy + 8y2

Checking for a perfect cube :

 9.2    2x4 + x3y - 4xy + 8y2  is not a perfect cube

Final result :

3y • (2x4 + x3y - 4xy + 8y2) ———————————————————————————— 4x


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