Canonical Law of Indices
The Canonical Law of Indices is the name used in Metopedia for the application of exponent laws under the Canonical Order of Operations. It treats powers, roots, grouping, and the sign of a term as parts of one interpretive system rather than as unrelated classroom rules.
The phrase is not a standard textbook title. It is a proposed title for a stricter form of the ordinary laws of indices, especially where notation leaves the base of an exponent unclear.
Definition
In canonical usage, an index law is not merely a rule for simplifying exponents. It is also a rule for identifying the base before simplification occurs. This means that x^2, -x^2, and (-x)^2 are not treated as interchangeable notational variants.
Canonical index rules
| Rule | Canonical form | Meaning |
|---|---|---|
| Implicit unity | x=x^1 | Every visible term carries an implicit first power. |
| Product of like bases | x^a x^b=x^{a+b} | Like bases combine by adding indices. |
| Quotient of like bases | \dfrac{x^a}{x^b}=x^{a-b} | Like bases divide by subtracting indices. |
| Power of a power | (x^a)^b=x^{ab} | Nested indices multiply. |
| Root as power | \sqrt[n]{x}=x^{\dfrac{1}{n}} | Roots are fractional indices. |
| Combined power and root | \sqrt[n]{x^m}=x^{\dfrac{m}{n}} | The numerator is the power and the denominator is the root. |
| Negative index | x^{-a}=\dfrac{1}{x^a} | A negative index denotes reciprocal form. |
Relationship to implicit unity
The Law of Implicit Unity gives ordinary terms a visible index before index laws are applied. For example:
This rule is central to COO because it prevents the exponent from silently extending beyond its intended base.
Relationship to roots
The Canonical Law of Indices treats roots as a subset of exponentiation. The radical symbol is therefore viewed as redundant notation rather than a distinct operation:
Sign and base identity
The proposed law requires the sign of a term to be distinguished from the base being exponentiated. This does not mean that negative values cannot be exponentiated. It means the notation must indicate whether the negative sign belongs to the base or operates outside the power.
| Expression | Canonical reading |
|---|---|
| -x^2 | -(x^2) |
| (-x)^2 | The grouped signed quantity is the base. |
| -(x)^2 | The sign is external to the squared base. |
Significance
The canonical formulation attempts to make the law of indices the controlling grammar for powers and roots. Its argument is that if roots are powers, and powers require bases, then notation must identify the base before any rule can be applied.