# Question #131a1

##### 1 Answer

Here's what I got.

#### Explanation:

Your starting expression looks like this

#12 sqrt(x-y)/y + 12 x/sqrt(x-y) - 12 sqrt((x-y)^2)/(x-y)#

The first thing to do here is identify the **common denominator** by taking a look at the three fractions you have in your expression.

Your goal when finding the common denominator is to get the denominators of the fractions to *match*, i.e. you need the three fractions to have **the same denominator**.

The first fraction has

In other words, the common denominator here will be the product of the three denominators. In this case, you have

#y * (x-y) * sqrt(x-y) -># thecommon denominator

To get to this point, multiply the first *fraction* by

#color(blue)(1) = ((x-y) * sqrt(x-y))/((x-y) * sqrt(x-y))#

to get

#12sqrt(x-y)/y * overbrace(((x-y) * sqrt(x-y))/((x-y) * sqrt(x-y)))^(color(blue)(=1)) = (12 * (x-y) * sqrt(x-y) * sqrt(x-y))/(y * (x-y) * sqrt(x-y))#

Notice that because you're essentially multiplying by **does not change**. You thus have

#12sqrt(x-y)/y = (12 * (x-y) * (sqrt(x-y))^2)/(y * (x-y) * sqrt(x-y))#

#= (12 * (x-y)(x-y))/(y * (x-y) * sqrt(x-y))#

Now do the same for the second fraction. You must multiply it by

#color(blue)(1) = (y * (x-y))/(y * (x-y))#

to get

#12x/sqrt(x-y) = 12x/sqrt(x-y) * overbrace((y * (x-y))/(y * (x-y)))^(color(blue)(=1))#

#= (12 * x * y * (x-y))/(y * (x-y) * sqrt(x-y))#

Finally, do the same for the third fraction. You must multiply it by

#color(blue)(1) = (y * sqrt(x-y))/(y * sqrt(x-y))#

to get

#12sqrt( (x-y)^3)/(x-y) = 12 sqrt((x-y)^3)/(x-y) * overbrace((y * sqrt(x-y))/(y * sqrt(x-y)))^(color(blue)(=1))#

#=(12y * sqrt( (x-y)^3 * (x-y)))/((y * (x-y) * sqrt(x-y)))#

#= (12y * (x-y)^2)/((y * (x-y) * sqrt(x-y)))#

Now you're ready to put all this together and simply the expression

#(12(x-y)^2)/((y * (x-y) * sqrt(x-y))) + (12 xy(x-y))/((y * (x-y) * sqrt(x-y))) - (12y(x-y)^2)/((y * (x-y) * sqrt(x-y)))#

Focus on the numerator first

#12(x-y)^2 + 12xy(x-y) - 12y(x-y)^2#

#12(x-y)[ (x-y) + xy - y(x-y)]#

#12(x-y)( x-y + color(red)(cancel(color(black)(xy))) - color(red)(cancel(color(black)(xy))) + y^2)#

#12(x-y)(y^2 + x - y)#

The expression now becomes

#(12(x-y)(y^2 + x - y))/(y(x-y)sqrt(x-y))#

Notice that the

#(12color(red)(cancel(color(black)((x-y))))(y^2 + x - y))/(y color(red)(cancel(color(black)((x-y)))) sqrt(x-y))#

to get

#color(green)(|bar(ul(color(white)(a/a)color(black)((12(y^2 + x - y))/(y * sqrt(x-y)))color(white)(a/a)|)))#

Note that

#sqrt(x-y) = (x-y)^(1/2)#

Because