Simple equation causing a lot of head ache

[Marin]

Hi there!

I was trying to solve an equation but got very perplexed by the fact that a certain number x=0 is both a solution and no solution:

Here´s the equation:

$$(x-a)[(x-a)^2+y^2+z^2]^{-3/2}+(x+a)[(x+a)^2+y^2+z^2]^{-3/2}=0$$

assume y=z=0:

Now, the equation becomes:

$$(x-a)[(x-a)^2]^{-3/2}+(x+a)[(x+a)^2]^{-3/2}=0$$

I guessed a solution at x=0

check: $$-a[(-a)^2]^{-3/2}+a[a^2]^{-3/2}=0$$, ok I assume it´s true

Now let´s use the exponent rule: (a^x)^y=a^(xy)

Then the equation becomes:

(*) $$(x-a)(x-a)^{-3}+(x+a)(x+a)^{-3}=0$$, or

$$(x-a)^{-2}+(x+a)^{-2}=0$$

Ok, plug once again x=0 and there coems the surprise:

$$(-a)^{-2}+(+a)^{-2}=1/a^2+1/a^2$$ is not equal to 0!

What is more, if you substitute x=0 before you contract the factors in (*) ypu come up with: $$(-a)^{-4}+(+a)^{-2}=1/a^4+1/a^2$$ again no 0, but also different from the result above!


I suppose I´m doing it somehow wrong woth the powers and exponents but I can´t figure it out!

If you see my mistake, please tell me!

Thanks in advance!

 

[tiny-tim]

Hi Marin!

… Now let´s use the exponent rule: (a^x)^y=a^(xy)

Sorry … doesn't always work, especially if y is a fraction and a is negative