SOLUTION: x to -2 power - y to -2 power divided by x to -1 power + y to -1 power (no parentheses in problem) answer is y - x divided by xy how do you get the answer

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: x to -2 power - y to -2 power divided by x to -1 power + y to -1 power (no parentheses in problem) answer is y - x divided by xy how do you get the answer      Log On


   

Question 558789: x to -2 power - y to -2 power divided by x to -1 power + y to -1 power
(no parentheses in problem)
answer is y - x divided by xy
how do you get the answer
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
As I understand your problem, you are to simplify the problem:
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%28x%5E%28-2%29-y%5E%28-2%29%29%2F%28x%5E%28-1%29%2B+y%5E%28-1%29%29
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Let's work on the numerator first by simplifying:
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%28x%5E%28-2%29-y%5E%28-2%29%29
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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%5E%28-2%29 to 1%2Fx%5E2
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and similarly we can convert:
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y%5E%28-2%29 to 1%2Fy%5E2
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Substituting these into the expression you were given for the numerator converts the numerator to:
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1%2Fx%5E2+-+1%2Fy%5E2
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The common denominator for this expression is x%5E2%2Ay%5E2. We can multiply the x-term by y%5E2%2Fy%5E2 to change it to y%5E2%2F%28x%5E2%2Ay%5E2%29. Similarly we can multiply the y-term by x%5E2%2Fx%5E2 to change it to x%5E2%2F%28x%5E2y%5E2%29.
Substituting these changed terms into the numerator makes it become:
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y%5E2%2F%28x%5E2%2Ay%5E2%29-x%5E2%2F%28x%5E2y%5E2%29 and finally combining the two terms over the common denominator results in the numerator becoming:
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%28y%5E2+-x%5E2%29%2F%28x%5E2%2Ay%5E2%29 <--- 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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%28x%5E%28-1%29%2B+y%5E%28-1%29%29
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We can convert these to positive exponential terms by placing the x%5E%28-1%29 under 1 to get 1%2Fx and the y%5E%28-1%29 under 1 to get 1%2Fy. This makes the denominator become:
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1%2Fx+%2B+1%2Fy
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The common denominator for this is x%2Ay 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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%28y%2Bx%29%2F%28x%2Ay%29 <--- 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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%28x%2Ay%29%2F%28y%2Bx%29
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Then multiplying this by the numerator the problem becomes:
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%28%28y%5E2+-x%5E2%29%2F%28x%5E2%2Ay%5E2%29%29%2A%28%28x%2Ay%29%2F%28y%2Bx%29%29
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Notice that the x%2Ay numerator of the second term can be canceled with the x%5E2%2Ay%5E2+ denominator of the first term to just an x%2Ay in the denominator and make the expression become:
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%28%28y%5E2+-x%5E2%29%2F%28x%2Ay%29%29%2A%28%281%29%2F%28y%2Bx%29%29
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Then notice that the y%5E2+-+x%5E2 is the difference of two squares. It can, therefore, be factored to %28y+-+x%29%2A%28y%2Bx%29 to convert the expression to:
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%28%28%28y+-x%29%2A%28y%2Bx%29%29%2F%28x%2Ay%29%29%2A%28%281%29%2F%28y%2Bx%29%29
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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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and as a final step, when you multiply everything out you end up with:
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%28y+-+x%29%2F%28x%2Ay%29
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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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