Yes. 3^^^4 < 4^^^4, and the denominator has the correct number and type of operations to be 3^^^4, since 3^^^4 = 3^^3^^3^^3 - however it’s actually smaller. Up arrow notation evaluates from right to left, which results in the largest possible number. The parentheses just muck things up and make the result smaller—so the answer is much much greater than 1.
3^^X is an exponential tower of 3s that is X high. 3^^^4 = 3^^3^^3^^3 (i.e. it’s 3 ^^ itself, with four 3s, just like 3^^4 is 3 ^ itself with four 3s). So 3^^^4 is an exponential tower of 3s that is 3^^3^^3 high. 3^^3^^3 is 3 ^^ itself 3 times, so it’s 3^^^3.
So 3^^^4 is an exponential tower of 3s 3^^^3 levels tall.
This is a rough guess, but since 3^^3 is a number I could type out in this comment box, and the ^^ operator is in general way less powerful than the ^^^ operator, I’d say yes.
Yes. 3^^^4 < 4^^^4, and the denominator has the correct number and type of operations to be 3^^^4, since 3^^^4 = 3^^3^^3^^3 - however it’s actually smaller. Up arrow notation evaluates from right to left, which results in the largest possible number. The parentheses just muck things up and make the result smaller—so the answer is much much greater than 1.
So 3^^^4 is an exponential tower of 3s 3^^4 levels tall?
Hm, let’s find out. (Open this if you want to have a go at finding that answer first)
3^^X is an exponential tower of 3s that is X high. 3^^^4 = 3^^3^^3^^3 (i.e. it’s 3 ^^ itself, with four 3s, just like 3^^4 is 3 ^ itself with four 3s). So 3^^^4 is an exponential tower of 3s that is 3^^3^^3 high. 3^^3^^3 is 3 ^^ itself 3 times, so it’s 3^^^3.
So 3^^^4 is an exponential tower of 3s 3^^^3 levels tall.
Thanks for the explanation. It was worth the brain explosion.
This is a rough guess, but since 3^^3 is a number I could type out in this comment box, and the ^^ operator is in general way less powerful than the ^^^ operator, I’d say yes.