When raising to a power and tetrating, which comes first? Is $^24^3$ equal to $(^24)^3$, or to $^2(4^3)$?

Question

I know tetration isn't quite an used operation, but anyway, what if it were featured in an expression?

For example, what does $^24^3$ mean? Is it $(^24)^3=(4^4)^3=4^{12}$ or is it $^2(4^3)=^2{64}=64^{64}$?

Answer 1

If I personally had to interpret it, I would rather it be $(^24)^3,$ so that tetration would bind more closely to the input than exponentiation does.

However, the real answer is: it's ambiguous. There is no general convention to determine an order of operations involving tetration, so if you use it, you ought to use parentheses to clarify your intent.

Answer 2

The only version I've seen of $ \;^h b^x $ was $$ a_h = \;^h b^x \implies \\\ a_0=x \\\ a_1=b^x \\\ a_2 = b^{a_1} = b^{b^x} = \;^2 b^x\\\ \vdots \\\ a_h = b^{a_{h-1}} = \underset{h \text{ times}}{\underbrace{ b^{ \cdots b^{b^x}}}} = \;^h b^x \\\ $$ Isn't this in wikipedia?

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p.s.: $(\;^24)^3 $ would be $ (4^4)^3 = 4^{3 \cdot 4} = 16777216 $
and $\;^2(4^3) $ would be $ (4^3)^{(4^3)} = 4^{3 \cdot 4^3 } = 394020061963944792... 6627990306816$ with 116 digits....
while my proposed interpretation would be $ 4^{(4^3)} =340282366920938463463374607431768211456$ having 39 digits

Answer 3

I would say it is $(^2 4)^3$. Similarly, to how multiplication is done before addition in the order of operations, exponentiation is done before multiplication. Therefore, I say that tetration is done before exponentiation.

However, just like in an expression with an order of evaluation that is hard to see, I would say to use parentheses to clarify what to do first.

Answer 4

Higher order operators are right-associative so I'd read it as $^{2}(4^3)$ which evaluates to $64^{64}$