Question 185459
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)



{{{(x+y)^(-2)}}} Start with the first expression.



{{{(1+1)^(-2)}}} Plug in {{{x=1}}} and {{{y=1}}}



{{{(2)^(-2)}}} Add



{{{1/(2^2)}}} Flip the fraction (to make the exponent positive)



{{{1/4}}} Square {{{2}}} to get {{{1/4}}}. 



So {{{(x+y)^(-2)=1/4}}} when {{{x=1}}} and {{{y=1}}}



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{{{x^(-2)+y^(-2)}}} Move onto the second expression



{{{1^(-2)+1^(-2)}}} Plug in {{{x=1}}} and {{{y=1}}} (note: these values cannot change now)



{{{1/1^2+1/1^2}}} Flip the fractions (to make the exponents positive)



{{{1/1+1/1}}} Square 1 to get 1



{{{1+1}}} Reduce



{{{2}}} Add



So {{{x^(-2)+y^(-2)=2}}} when {{{x=1}}} and {{{y=1}}}



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Since {{{(x+y)^(-2)=1/4}}} and {{{x^(-2)+y^(-2)=2}}} when {{{x=1}}} and {{{y=1}}}, this means that {{{(x+y)^(-2)<>x^(-2)+y^(-2)}}}