Simplify:  `(5+3/(x+y))/( 3+x/(x^3 y))`   5+3/x+y ---------- 3+x/x^3y

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durbanville's profile pic

durbanville | High School Teacher | (Level 2) Educator Emeritus

Posted on

`(5+3/(x+y))/ (3+x/(x^3y))`

Simplify the top and simplify the bottom separately first. The LCD for the top is (x+y):

`5+3/(x+y) = (5(x+y)+3)/(x+y)`

`= (5x + 5y +3)/ (x+y)`

Now simplify the bottom which has an LCD of `x^3y` :

`3+x/(x^3y)= (3(x^3y) + x)/(x^3y)`

Now put the two together. It is esier to use a traditional division symbol( `divide` ) when doing fractions over fractions:

`((5x+5y +3)/(x+y)) divide ((3x^3y + x)/(x^3y))` 

The rules of fractions tell us to use the recipricol when diciding fractions so we change the divide symbol to times (x) and swap the second term around:

`(5x+5y+3)/(x+y) times (x^3y)/(3x^3y +x)`

As this is a multiplication and we have no like terms to cross cancel, multiply the top and bottom:

`= ((x^3y)(5x+5y+3))/((x+y)(3x^3 y+x))`

Simplify the second bracket at the bottom:

`= ((x^3y)(5x+5y +3))/( (x+y)(x)(3x^2y+1))`

Cross cancel one of the x-es from `x^3y` witth the (x) at the bottom:

`((x^2y)(5x+5y+3))/((x+y)(3x^2y+1))`

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pramodpandey's profile pic

pramodpandey | College Teacher | (Level 3) Valedictorian

Posted on

`5+3/(x+y)=5/1+3/(x+y)=5(x+y)+3/(x+y)=(5x+5y+3)/(x+y)`

`3+x/(x^3y)=3/1+1/(x^2y)=(3x^2y+1)/(x^2y)`

`(5+3/(x+y))/(3+x/(x^3y))=((5x+5y+3)/(x+y))/((3x^2y+1)/(x^2y))`

`=((5x+5y+3)(x^2y))/((x+y)(3x^2y+1))`

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