[00:00] we have three completely different types of notation to write the same fact? position of the 3 over the 2 that indicates the operation.
[00:13] which is the same fact, you introduce this new squiggly radical symbol. you write out a word for the operation. This weird discrepancy in notation isn't just counterintuitive,
[00:28] Rather than making seemingly different facts look the same, which is what math should do, it makes three facts that should obviously be the same look artificially different. stack exchange for a more symmetric notation here.
[00:44] In our example, the way this would work is you write a triangle with a 2 in the lower left, a 3 on the top, and an 8 on the lower right. and the symbol as a whole represents the value that should go in that missing corner.
[01:00] To express log base 2 of 8, which is asking the question 2 to the what equals 8, Again, the symbol as a whole represents the value that should go in that missing corner.
[01:12] If you want to express the cube root of 8, you remove the lower left corner. value that should go in the missing corner. This much more clearly expresses the relationship of all three operations.
[01:25] The definition alone is mildly pleasing, but where it becomes useful is in seeing how the rules for exponentiation logs and radicals are all really the same. The most extreme example might be how with our current notation, there are six,
[01:39] and it looks like a complete mess. operations follow the same basic aesthetic pattern.
[01:51] Admittedly, it looks a little bit weird when your brain has already been trained with the traditional notation, but our brains are really good at picking up on patterns. you only need to remember one pattern which unlocks all six of these cases.
[02:06] Essentially, any rule that's associated with exponents logs and radicals becomes essentially three times faster to learn and to recognize. moment and mess around with what some of these rules for exponents look like.
[02:20] but it's super satisfying once it all clicks.