---
title: 'The Triangle of Power'
source: 'https://youtube.com/watch?v=JNVZK4ofW-Y'
video_id: 'JNVZK4ofW-Y'
date: 2026-08-08
duration_sec: 146
---

# The Triangle of Power

> Source: [The Triangle of Power](https://youtube.com/watch?v=JNVZK4ofW-Y)

## Summary

This video explores the inconsistent notation used for exponentiation, logarithms, and roots, and proposes a more symmetric alternative called the 'Triangle of Power.' The presenter argues that this triangular notation better reflects the underlying relationships between these operations, making rules easier to learn and recognize.

### Key Points

- **Inconsistent Notation Problem** [00:00] — The video highlights that exponentiation, logarithms, and roots use three completely different notations (e.g., 2^3, log₂8, ∛8) to represent the same underlying relationship, which is counterintuitive and makes related facts look artificially different.
- **Triangle of Power Concept** [00:28] — A more symmetric notation is proposed: a triangle with the base (e.g., 2) in the lower left, the exponent (e.g., 3) on top, and the result (e.g., 8) in the lower right. The symbol as a whole represents the missing value.
- **Expressing Logs and Roots** [01:00] — Log base 2 of 8 is expressed by removing the top corner (asking 2 to what equals 8), and the cube root of 8 is expressed by removing the lower left corner. This clearly shows the relationship between all three operations.
- **Unifying Rules** [01:25] — The notation makes rules for exponentiation, logs, and radicals appear as the same pattern. The presenter notes that with traditional notation, there are six cases that look like a mess, but with the triangle, they follow one aesthetic pattern.
- **Learning Efficiency** [02:06] — Using the Triangle of Power, any rule associated with exponents, logs, and radicals becomes three times faster to learn and recognize, as only one pattern needs to be remembered to unlock all six cases.

### Conclusion

The Triangle of Power offers a more intuitive and unified notation for exponentiation, logarithms, and roots, making mathematical relationships clearer and rules easier to learn. While it may look unfamiliar at first, it provides a satisfying and efficient way to understand these operations.

## Transcript

we have three completely different types of notation to write the same fact? position of the 3 over the 2 that indicates the operation.
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,
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.
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.
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.
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.
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,
and it looks like a complete mess. operations follow the same basic aesthetic pattern.
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.
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.
but it's super satisfying once it all clicks.