Question 72429
To simplify this expression, you can first note that {{{x^2 - y^2}}} is the difference
of two squares.  Therefore, it factors into {{{(x - y)* (x + y)}}}.  Substitute these and
the problem becomes:
.
{{{ (5)/(x+y) + (5)/((x-y)*(x+y))}}}
.
Notice that the first fraction lacks a (x-y) term in the denominator.  If it had one, then
this first fraction could be combined with the second fraction because they both would have
the same denominator.
.
Let's multiply the first fraction by {{{(x-y)/(x-y)}}}. Because the numerator is identical to
the denominator in this multiplier, it is equivalent to multiplying by 1.  The multiplication
results in:
.
{{{ ((5)/(x+y))*((x-y)/(x-y)) + (5)/((x-y)*(x+y))}}}
.
Since the two fractions now have a common denominator of {{{(x-y)*(x+y)}}} their numerators
may be put over this common denominator to get:
.
{{{(5*(x-y)+5)/((x-y)*(x+y))}}}
.
Then, doing the multiplication in the numerator results in:
.
{{{(5*x - 5*y + 5)/((x-y)*(x+y))}}}
.
And factoring the common 5 in the numerator:
.
{{{5*(x - y +1)/((x-y)*(x+y))}}}
.
Hope this gives you some additional insight into working with fractional expressions.