Question 558789
As I understand your problem, you are to simplify the problem:
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{{{(x^(-2)-y^(-2))/(x^(-1)+ y^(-1))}}}
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Let's work on the numerator first by simplifying:
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{{{(x^(-2)-y^(-2))}}}
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From the rules of exponents, if you have a negative exponent you can convert that by dividing 1 by the same term with a positive exponent.  Following that rule we can convert:
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{{{x^(-2)}}} to {{{1/x^2}}}
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and similarly we can convert:
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{{{y^(-2)}}} to {{{1/y^2}}}
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Substituting these into the expression you were given for the numerator converts the numerator to:
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{{{1/x^2 - 1/y^2}}}
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The common denominator for this expression is {{{x^2*y^2}}}. We can multiply the x-term by {{{y^2/y^2}}} to change it to {{{y^2/(x^2*y^2)}}}. Similarly we can multiply the y-term by {{{x^2/x^2}}} to change it to {{{x^2/(x^2y^2)}}}.
Substituting these changed terms into the numerator makes it become:
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{{{y^2/(x^2*y^2)-x^2/(x^2y^2)}}} and finally combining the two terms over the common denominator results in the numerator becoming:
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{{{(y^2 -x^2)/(x^2*y^2)}}} <--- remember this numerator for later
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Now let's do a similar procedure for the denominator of the problem. We had a denominator of:
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{{{(x^(-1)+ y^(-1))}}}
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We can convert these to positive exponential terms by placing the {{{x^(-1)}}} under 1 to get {{{1/x}}} and the {{{y^(-1)}}} under 1 to get {{{1/y}}}. This makes the denominator become:
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{{{1/x + 1/y}}}
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The common denominator for this is {{{x*y}}} and when we place the two terms over this common denominator (by following the general process we used for the numerator) the result is:
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{{{(y+x)/(x*y)}}} <--- this is now the denominator for the original problem
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We are now going to divide this denominator into the numerator that we remembered for later. 
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The denominator is a fraction. Remember from long ago that you can divide by a fraction by inverting it and multiplying. So we can divide by inverting the denominator and multiplying it by the numerator we remembered. When we invert the denominator we get:
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{{{(x*y)/(y+x)}}}
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Then multiplying this by the numerator the problem becomes:
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{{{((y^2 -x^2)/(x^2*y^2))*((x*y)/(y+x))}}}
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Notice that the {{{x*y}}} numerator of the second term can be canceled with the {{{x^2*y^2 }}} denominator of the first term to just an {{{x*y}}} in the denominator and make the expression become:
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{{{((y^2 -x^2)/(x*y))*((1)/(y+x))}}}
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Then notice that the {{{y^2 - x^2}}} is the difference of two squares. It can, therefore, be factored to {{{(y - x)*(y+x)}}} to convert the expression to:
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{{{(((y -x)*(y+x))/(x*y))*((1)/(y+x))}}}
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Next notice and cross out the common factor in the numerator with a common factor in the denominator:
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{{{(((y -x)*(cross(y+x)))/(x*y))*((1)/cross(y+x))}}}
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and as a final step, when you multiply everything out you end up with:
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{{{(y - x)/(x*y)}}}
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which is just as you said it should be.
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I hope that with a little careful tracking you can work your way through this and that it gives you some insight to the problem. Good luck ...
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