Canonical notation

Canonical notation is the expression style used by the Canonical Order of Operations to replace radical notation with explicit exponent notation. Its purpose is to show powers, roots, reciprocal roots, and negative roots through a single exponent structure.

Root conversion

The core rule is that roots should be written as fractional exponents:

Radical notation	Canonical notation
$\sqrt{a}$	$a^{\frac{1}{2}}$
$\sqrt[3]{a}$	$a^{\frac{1}{3}}$
$\sqrt[n]{a}$	$a^{\frac{1}{n}}$
$\sqrt[n]{a^{m}}$	$a^{\frac{m}{n}}$

In MediaWiki math markup, canonical pages should use \frac{...}{...}, not slash-style exponents, for displayed fractional powers.

Reciprocal roots

Reciprocal roots are written with negative fractional exponents:

Radical notation	Canonical notation
$\frac{1}{\sqrt{a}}$	$a^{- \frac{1}{2}}$
$\frac{1}{\sqrt[3]{a}}$	$a^{- \frac{1}{3}}$
$\frac{1}{\sqrt[n]{a}}$	$a^{- \frac{1}{n}}$
$\frac{1}{\sqrt[n]{a^{m}}}$	$a^{- \frac{m}{n}}$

Negative reciprocal roots

Negative reciprocal roots keep the negative sign outside the exponentiated expression:

Radical notation	Canonical notation
$- \frac{1}{\sqrt{a}}$	$- a^{- \frac{1}{2}}$
$- \frac{1}{\sqrt[3]{a}}$	$- a^{- \frac{1}{3}}$
$- \frac{1}{\sqrt[n]{a}}$	$- a^{- \frac{1}{n}}$

Powers of roots

Powers of roots are converted by multiplying exponents:

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

This rule follows the power-of-a-power structure:

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

Purpose

Canonical notation is intended to remove ambiguity caused by the radical symbol and to make the exponent structure visible. It is one of the less disputed surface features of COO when used only as notation, but it becomes part of a stronger framework when combined with the Law of Implicit Unity and COO's treatment of negative bases.

Root conversion

Reciprocal roots

Negative reciprocal roots

Powers of roots

Purpose

See also