What Kind of Math Is This?! 😳

Added: 2026-05-05

[00:00:00]Hello my lovelies. It's Susanna and today I want to show you something that I never came across during my math studies. Not at school, not at university. It is this expression here that we just have to calculate.

[00:00:16]But what is it? It looks pretty similar to two to the power of four. But this here is definitely something different than this thing here because here we have the four on the right side and not on the left side. But just to repeat it again, what is 2 to the^ of 4? It means that we take the base the two and multiply it by itself. So 2 * 2 * 2 * how often until we see four of these twos here. So one more time. So exponentiation is the repetition of multiplication.

[00:01:08]Okay. But what is this here then? This is the so-called tetration.

[00:01:14]It's also called the fourth tetration of two. Sometimes also noted as two double up arrow four.

[00:01:28]And it's not the repetition of multiplication like in this case but the repetition of exponentiation.

[00:01:39]So we take the base and instead of times 2 * 2 we raise this number now to the power of itself again. So we have 2 to the power of two to the power of two how often until we see four of these twos.

[00:02:03]So one last time and before we calculate this here now just for you to see these different levels this here is the repetition of exponentiation.

[00:02:17]This here is the repetition of multiplication and then there is another level where you have the repetition of addition.

[00:02:28]So if you have 2 + 2 + 2 + 2 then you can also write this as 4 * 2 which means here the first level is the multiplication which is the repetition of addition. The next level is exponentiation which is the repetition of multiplication. then and the next level is the tetration where we have the repetition of exponentiation.

[00:03:02]Okay. But now let's calculate this beautiful power tower that we have here.

[00:03:08]The rule is that you always have to start from the right side. So if there are no parentheses in here, usually you might see parentheses, then you have to start with the parenthesis. But if there are no parentheses at all, the rule is to start from the right. So we have to start with 2 to the^ of two here first.

[00:03:32]So that we have this here 2 to the^ of two. But 2 to the^ of two is equal to 4.

[00:03:40]Now an smaller power tower. Same rule.

[00:03:44]We have to keep going from the right side. So 2 to the^ of 4 first which means we have two and 2 to the^ 4 ah perfect we already had it here. So this is equal to 16. So that we have 2 to the^ of 16 which is a pretty large number. It is 65,536.

[00:04:13]So you can see that this result of the fourth tetration of two is way bigger than this result here from two to the power of four. So it's definitely not the same. But I also want to show you what would happen at this step here with this power tower if you didn't know the rule that you would have to start from the right side. So let's take this power tower and let's see what happens if you just don't know about this rule and just start from the left side. So as if there were parentheses here you would start with 2 to the^ of two which is equal to four and then you would still have to the power of two and to the power of two. Then you have a new power tower, but you're like, I start from the left side. So I have parentheses here. I start with 4 to the^ of two, which equals 16. And then I only have all of this raised to the power of two left.

[00:05:23]And 16^ squared equals 256.

[00:05:28]And you can see this is definitely not the same result. So this is not correct.

[00:05:36]You always have to start from the right side and it really makes a different if you don't do this. Okay, let's get used to this notation with different examples. Something like three tetrated to two. What does it mean? Now we know that we take the base the three and raise it to the power of itself. So raised to the power of three how often until we have two of these threes. H we are already done with this. So this is a small power tower. It's just a normal power. uh 3 to the^ 3 is 3 * 3 * 3 which is equal to 27.

[00:06:26]And the last example if we switch these two numbers. So if we have two tetrated to three then this means we take the base we two the two and raise it to the power of itself raised to the power of two raised to the power of two until we see three of these twos. And now we start from the right side. We know that now. So we have to start here so that we keep our base. 2 ^ of 2 = 4. 2 to the^ of 4. We already had that. This was 16.

[00:07:06]I hope you learned something new today.

[00:07:08]Let me know in the comments if you've heard of tetration before. I wish you a wonderful day and I hope to see you in one of my next videos. Take care.