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So, assume that we need to solve for x for a given equation,

$2^{3^{4^x}} = 516$

Now, here $x = {1}/{2}$ (when going down to up, by factorising $512$, and equating the powers) but what if we went up to down instead of down to up? meaning we do $2^3$ first, instead of $4^x$ , where $x = 1/2$

I know this sounds like a pathetic baby question, but just a small doubt I had in the back of my peanut sized brain. if possible, also show as to WHY we go down to up instead of it being reversed.

thanks :)

miracle173
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Dhruva Choudhary
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  • See Henning Makholm's answer to the linked question in particular. As you observed $a^{(b^c)}$ is different from $(a^b)^c$. So we need to pick one of those to be the meaning of $a^{b^c}$. As we (assume integers only to avoid complications) have $$(a^b)^c=a^{b\cdot c},$$ we have a way of rewriting the latter alternative more simply already. Therefore the former meaning is standard. May be better to call it a widely used convention? Anyway, in calculus and probability you see a lot of expressions like $$e^{-x^2/2},$$ where it is imperative that the exponent is evaluated first. This fits. – Jyrki Lahtonen Dec 22 '24 at 08:17
  • Thanks @JyrkiLahtonen – Dhruva Choudhary Jun 03 '25 at 12:33

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