# Simplify:( x/x^2-16 -1/x-4)/4/x+4 The Quotient of Two Rational Expressions

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[x/(x^2 - 16) - 1/(x-4) ] / [4/(x+4)]

First let us simplify:

We know that x^2 - 16 = (x-4)(x+4)

==> [ (x - 1(x+4)/ (x^2 - 16) ] / [4/(x+4)]

==> [ -4/(x^2 - 16) ] * (x+4)/4

==> -4*(x+4) /4(x^2 - 16)

==> -**1/(x-4)**

To simplify the expression you have provided first write it with proper brackets so that you do not confused. This can be done like this: [x/(x^2 - 16) - 1/(x-4) ] / [4/(x+4)]

Now,

[x/(x^2 - 16) - 1/(x-4) ] / [4/(x+4)]

=[x/(x-4)(x+4) - 1/(x-4) ] / [4/(x+4)]

=[{x-(x+4)}/(x+4)(x-4)]/[4*(x-4)/(x+4)(x-4)]

(x+4)(x-4) get cancelled in the numerator and denominator.

We are left with -4/[4*(x-4)]= **-1/(x-4)**

Simplify:( x/x^2-16 -1/x-4)/4/x+4.

Solution:

{x/(x^2-16) - 1/(x-4)}/ (4/(x+4)) or

{[x/(x^2-16-1)/(x-4)]/4 }/x and then add .

Such cases arise as there is no unique representation by using sufficient brackets.

I take the 1st choice of freedom of putting bracket.

(a/b)/(c/d) = ad/bc.

x/(x^2-16) - 1/(x-4) = [x- (x+4)]/(x^2-16), as x^-16 is made the common denominator.

=4/(x^2-16).

So {x/x^2-16)-1/(x-4)}/{4/(x+4)] = -4 (x+4)/[(x^2-16)4

{x/x^2-16)-1/(x-4)}/{4/(x+4)]= -(x+4)/(x^2-16)

{x/x^2-16)-1/(x-4)}/{4/(x+4)]= -(x+4)/(x-4)(x+4)

{x/x^2-16)-1/(x-4)}/{4/(x+4)] = 1/(x-4)