how to show that (x + y)^-2 and x^-2 + y^2 are not equivalent

Question

 

I would think they are not equivalent because part of the equation is in paranthesis.  Is this correct?

Answer 1

First of all, let's look at the exponent.  a^-2 means that the value of a is reciprocated and squared;  That means that a^-2 = 1/a^2.  Therefore, (x+y)^-2 is 1/(x+y)^2.

 

Now, (x+y)^2 means (x+y)(x+y) or x^2 + 2xy + y^2.  (There is a shortcut for that).

 

So, 1/(x^2 + 2xy + y^2) does not equal (1/x^2) + y^2.  (Note the parentheses)

 

If we rewrite this last expression as a fraction, it becomes (1/x^2) +  (x^2 y^2)/x^2,  or (1 + x^2y^2)/x^2.

 

I hope this helps.

 

Kevin

Answer 2

Try their values for x=1, y=2. They are different.  1/9 and 9 1/2 respectively.