power question

[Levido]

Hi everyone,

I've got a question about powers. I got the answer right instinctively but wanted help if someone can show me why the method they give in the solutions works.

Question:
Find the value of x >0 that satisfies the equation:



Solution:
Eliminating one of the x’s would not have any effect on the end result, because of the nature of infinity. Therefore, by removing the first x, we find that all those remaining in the tower of x’s must also equal 2. This then permits us to rewrite this equation as $x^2=2$

I'll say that the answer makes sense to me, but I don't get the "This then permits us to rewrite this equation as $x^2=2$". If you do this an "almost infinite amount" of times you end up with $x^x=2$, obviously this is not true. Maybe I missed a lesson on limits in pre-Calculus?

Thanks,
Levid0

 

[blamocur]

Maybe using parens might help:
$$x^{x^{x^{x^{...}}}} = x^{\left(x^{x^{x^{...}}}\right)}$$I.e., the expression in the parens is "as inifinite" as the left hand side, thus its value must also be 2.

 

[Levido]

Maybe using parens might help:
$$x^{x^{x^{x^{...}}}} = x^{\left(x^{x^{x^{...}}}\right)}$$I.e., the expression in the parens is "as inifinite" as the left hand side, thus its value must also be 2.

I really can't see it still. What do you mean by "As infinite"?

Thanks for your reply

 

[blamocur]

I really can't see it still. What do you mean by "As infinite"?

Thanks for your reply

I meant to say that the expression in the parenthesis (in the right hand side) must have the same value as the expression in the left hand side. I.e., removing one "x" does not change the value of this infinite expression.

 

[Levido]

I meant to say that the expression in the parenthesis (in the right hand side) must have the same value as the expression in the left hand side. I.e., removing one "x" does not change the value of this infinite expression.

Oh, I see it now. Thanks for your help Blamocur