SOLUTION: Adding and Subtracting Radical Expressions: Perform the operation and Simplify. {{{ (5)/(x+y) }}} + {{{ (5)/(x^2-y^2) }}}

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Question 72429: Adding and Subtracting Radical Expressions:
Perform the operation and Simplify.
+%285%29%2F%28x%2By%29+ + +%285%29%2F%28x%5E2-y%5E2%29+
Found 2 solutions by Cintchr, bucky:
Answer by Cintchr(481) About Me  (Show Source):
You can put this solution on YOUR website!
You must have common denominators.
+%285%29%2F%28x%2By%29+ + +%285%29%2F%28x%5E2-y%5E2%29+
Factor the second denominator
+%285%29%2F%28x%2By%29+ + +%285%29%2F%28%28x-y%29%28x%2By%29%29+
Multiply the first fraction by (x-y)/(x-y)
+%28%285%29%28x-y%29%29%2F%28%28x%2By%29%28x-y%29%29+ + +%285%29%2F%28%28x-y%29%28x%2By%29%29+
Distribute the 5 in the first numerator
+%28%285x-5y%29%29%2F%28%28x%2By%29%28x-y%29%29+ + +%285%29%2F%28%28x-y%29%28x%2By%29%29+
Combine the two fractions by adding the numerators
+%285x-5y%2B5%29%2F%28%28x%2By%29%28x-y%29%29+
You can factor out a 5 from the numerator if your instructor wants you so go that far
+%285%28x-y%2B1%29%29%2F%28%28x%2By%29%28x-y%29%29+

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
To simplify this expression, you can first note that x%5E2+-+y%5E2 is the difference
of two squares. Therefore, it factors into %28x+-+y%29%2A+%28x+%2B+y%29. Substitute these and
the problem becomes:
.
+%285%29%2F%28x%2By%29+%2B+%285%29%2F%28%28x-y%29%2A%28x%2By%29%29
.
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 %28x-y%29%2F%28x-y%29. Because the numerator is identical to
the denominator in this multiplier, it is equivalent to multiplying by 1. The multiplication
results in:
.

.
Since the two fractions now have a common denominator of %28x-y%29%2A%28x%2By%29 their numerators
may be put over this common denominator to get:
.
%285%2A%28x-y%29%2B5%29%2F%28%28x-y%29%2A%28x%2By%29%29
.
Then, doing the multiplication in the numerator results in:
.
%285%2Ax+-+5%2Ay+%2B+5%29%2F%28%28x-y%29%2A%28x%2By%29%29
.
And factoring the common 5 in the numerator:
.
5%2A%28x+-+y+%2B1%29%2F%28%28x-y%29%2A%28x%2By%29%29
.
Hope this gives you some additional insight into working with fractional expressions.