The Canonical Laws of Indices

This article lists the Canonical Order of Operations index laws. Law XV is the Law of Implicit Unity.

The Canonical Laws of Indices are the exponent and index rules used by Andrew Lehti's Canonical Order of Operations. The first fourteen laws restate or adapt standard exponent principles, while Law XV formalizes the Canonical framework's distinctive rule: the Law of Implicit Unity.

I. Implicit power of one

Every number or variable is implicitly raised to the first power when no exponent is written:

$a=a^1$

This law establishes a base as an exponent-bearing term even when the exponent is not visible.

II. Multiplying powers with the same base

When multiplying powers with the same base, add the exponents:

$a^m\times a^n=a^{m+n}$

III. Dividing powers with the same base

When dividing powers with the same base, subtract the exponent in the denominator:

$\frac{a^m}{a^n}=a^{m-n}$

where $a\ne 0$.

IV. Power of a power

When raising an exponentiated expression to another power, multiply the exponents:

$(a^m{)}^n=a^{m\times n}=a^{mn}$

V. Parenthesized base raised to a power

A parenthesized base carries an implicit first power:

$(a{)}^m=(a^1{)}^m=a^{1\times m}=a^m$

VI. Zero exponent law

Any nonzero base raised to zero equals one:

$a^0=1$

where $a\ne 0$.

VII. Negative exponent law

A negative exponent expresses the reciprocal of the positive exponent:

$a^{-m}=\frac{1}{a^m}$

where $a\ne 0$.

VIII. Distributing a power over multiplication

A power distributes over a product:

$(ab{)}^m=a^m\times b^m$

IX. Distributing a power over division

A power distributes over a quotient:

${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$

where $b\ne 0$.

X. Power as inverse root interpretation

A unit fractional exponent corresponds to an indexed root:

$a^{\frac{1}{n}}=\sqrt[n]{a}$

XI. Fractional exponent representing power and root

A fractional exponent combines a power and a root:

$a^{\frac{m}{n}}=\sqrt[n]{a^m}$

In this interpretation, the numerator is the power and the denominator is the root index.

XII. Power of one

Any base raised to the first power remains itself:

$a^1=a$

XIII. Zero to a positive exponent

Zero raised to a positive exponent equals zero:

$0^m=0$

where $m>0$.

XIV. Interpreting negative signs with powers

In the Canonical framework, the negative sign remains external when a negative base is interpreted through implicit unity:

$(-a{)}^m=(-a^1{)}^m=-a^{1\times m}=-a^m$

This law is the bridge between ordinary index handling and Canonical negative-base interpretation.

XV. Law of Implicit Unity

The Law of Implicit Unity states that every base, whether written plainly or inside parentheses, is implicitly raised to the first power before another exponent is applied. This permits parentheses around a base to be removed through index-law multiplication rather than through repeated-multiplication substitution.

For a positive or unsigned base:

$(a{)}^m=(a^1{)}^m=a^{1\times m}=a^m$

For a negative base in the Canonical framework:

$(-a{)}^m=(-a^1{)}^m=-a^{1\times m}=-a^m$

Root rewrites under the laws

The Canonical framework uses fractional exponents instead of slash notation or radical-first notation:

$\sqrt{a}=a^{\frac{1}{2}}$

$\sqrt[3]{a}=a^{\frac{1}{3}}$

$\sqrt[n]{a^m}=a^{\frac{m}{n}}$

$\frac{1}{\sqrt[n]{a}}=a^{-\frac{1}{n}}$

$-\frac{1}{\sqrt[n]{a}}=-a^{-\frac{1}{n}}$

$(\sqrt[n]{a}{)}^m=a^{\frac{m}{n}}$

See also

• Canonical Order of Operations
• Law of Implicit Unity
• Canonical notation
• Argument for the Removal of the Radical Symbol
• History of Negative Squares