Express in simplest form

Question

10x^2y       (x+y)^2     x^2-y^2

---------  •  ----------- ÷ -----------

x^2+xy          2x            5y^2

Answer 1

The first thing to do here is to factor everything and recognize that we are dividing by the last term.  Recall that when you divide by a fraction, it is the same as multiplying by the reciprocal of that fraction.

 

So,

10x²y  •  (x+y)2   •   5y²

x²+xy      2x             x²-y²

Notice I've flipped the last term and am multiplying rather than dividing.

 

Now let's factor each term.

10x²y  •  (x+y)²  •       5y²
x(x+y)        2x      (x+y)(x-y)

 

Removing the bolded terms, we have

10x²y  5y²
x 2x (x-y)  

 

Continuing in this fashion

10x²y 5y²
x 2x (x-y)

 

Removing the marked x's

10y 5y²

2 (x-y)

 

Since 10/2 = 5, we are left with

5y 5y2 = 25y³
(x-y)       (x-y)

 

Hope this has been of help.

 

Answer 2

This is just a matter of (1) cancelling and (2) factoring.  As we progress, the expression gets simpler -- until it is in simplest form.

 

Cancelling and factoring may be done in several different ways (some will be faster and some simpler), but it won't change the result.  So, let's go!

 

10x^2y       (x+y)^2     x^2-y^2
---------  •  ----------- ÷ -----------
x^2+xy          2x            5y^2

 

To divide by a fraction, invert and multiply.

10x^2y       (x+y)^2       5y^2 
---------  •  ----------- • -----------
x^2+xy          2x            x^2-y^2

 

Cancel 10x^2y and 2x

5xy            (x+y)^2       5y^2 
---------  •  ----------- • -----------
x^2+xy          1            x^2-y^2

 

Factor an x out of the first denominator and cancel with the numerator.

5y              (x+y)^2       5y^2 
---------  •  ----------- • -----------
x+y                1            x^2-y^2

 

Cancel an (x+y):

5y              (x+y)         5y^2 
---------  •  ----------- • -----------
  1                 1            x^2-y^2

 

Realize that  (x^2-y^2) is  (x+y)(x-y)

5y              (x+y)         5y^2 
---------  •  ----------- • -----------
  1                 1            (x+y)(x-y)

 

Cancel an (x+y):

5y                 1              5y^2 
---------  •  ----------- • -----------
  1                 1              (x-y)

 

When you get to the end, stop  (as the Mad Hatter and the rabbit said to Alice).

25y^3

-------

(x-y)