[Hypertype Theory](https://beta.observablehq.com/@ishi/introduction-to-hypertype-theory) makes the radical suggestion that a commutative power operator would be preferable to the traditional non-commutative power operator $a^b$. Are there any equations in physics that would be simplified by using such an operator?

**Non-commutative Power Operator**: $a^b = e^{\lg(a)×b}$

**Commutative Power Operator**: $a \# b = b \# a = e^{\lg(a)×\lg(b)} = a^{\lg(b)} = b^{\lg(a)}$

**Identity Element**: $a\#e = e\#a = a$

**Inverse Operator**: $a \backslash b = e^{\frac{\lg(a)}{\lg(b)}} = a^{\frac{1}{\lg(b)}}$

**Traditional Connection**: $a^b = a\#e^b$

**Terminology**:

The $\#$ operation is called *expansion* and $a \# b$ reads *“a expands b”*.

The $\backslash$ operation is called *contraction* and $a \backslash b$ reads *“a contracts b”*.

Obviously, many equations like $e = mc^2$ would suffer from such a replacement and should be left untouched. The question is interested in more complex equations that could be simplified.

**Note**: This is important because addition and multiplication are commutative, but exponentiation is not. If one thinks of a multiplication as doing multiple additions and an exponentiation as doing multiple multiplications, the lack of commutativity for exponentiation is bothersome. Instead, one should think of addition, multiplication, and expansion as being the same operation applied to different exponential scales. Obviously, the same goes for subtraction, division, and contraction.

**Algebraic Foundations**: [Hypertype Theory](https://beta.observablehq.com/@ishi/introduction-to-hypertype-theory) shows that an infinite number of such pairs of operators can be created in a recursive manner using very simple equations. The simplicity and beauty of the model suggests that we are looking at something interesting.