SOLUTION: Show by counterexample that (x+y)^-2 and x^-2+y^-2 are not equivalent. please help

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Question 185459: Show by counterexample that (x+y)^-2 and x^-2+y^-2 are not equivalent. please help
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Proof by Counterexample:


Simply pick two random values for "x" and "y" and plug them in. I'm going to use x=1 and y=1 (these values are small and nonzero)


%28x%2By%29%5E%28-2%29 Start with the first expression.


%281%2B1%29%5E%28-2%29 Plug in x=1 and y=1


%282%29%5E%28-2%29 Add


1%2F%282%5E2%29 Flip the fraction (to make the exponent positive)


1%2F4 Square 2 to get 1%2F4.


So %28x%2By%29%5E%28-2%29=1%2F4 when x=1 and y=1


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x%5E%28-2%29%2By%5E%28-2%29 Move onto the second expression


1%5E%28-2%29%2B1%5E%28-2%29 Plug in x=1 and y=1 (note: these values cannot change now)


1%2F1%5E2%2B1%2F1%5E2 Flip the fractions (to make the exponents positive)


1%2F1%2B1%2F1 Square 1 to get 1


1%2B1 Reduce


2 Add


So x%5E%28-2%29%2By%5E%28-2%29=2 when x=1 and y=1


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Since %28x%2By%29%5E%28-2%29=1%2F4 and x%5E%28-2%29%2By%5E%28-2%29=2 when x=1 and y=1, this means that %28x%2By%29%5E%28-2%29%3C%3Ex%5E%28-2%29%2By%5E%28-2%29