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

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 $$^2(4^3)=^2{64}=64^{64}$$?

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.

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?

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

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.