This page collects examples of rewrites used in the Canonical Order of Operations. The aim is not to show every possible calculation, but to show how COO makes the base, sign, exponent, and root structure explicit.
Implicit unity
| Starting form | Canonical rewrite |
|---|
| x | x^1 |
| 7 | 7^1 |
| (x) | (x^1) |
| (ab) | (a^1b^1) |
Radical removal
| Starting form | Canonical rewrite |
|---|
| \sqrt{x} | x^{\dfrac{1}{2}} |
| \sqrt[3]{x} | x^{\dfrac{1}{3}} |
| \sqrt{x^2} | x^{\dfrac{2}{2}} |
| \sqrt[5]{x^3} | x^{\dfrac{3}{5}} |
| \dfrac{1}{\sqrt{x}} | x^{-\dfrac{1}{2}} |
Power of a power
| Starting form | Canonical rewrite |
|---|
| (x^2)^3 | x^6 |
| (x^{\dfrac{1}{2}})^2 | x^1 |
| (x^{\dfrac{m}{n}})^p | x^{\dfrac{mp}{n}} |
Sign placement
| Starting form | Canonical reading |
|---|
| -x^2 | -(x^2) |
| (-x)^2 | The signed grouped term is the base. |
| -(x^2) | The negative sign is explicit and external. |
| (-1)x^2 | The sign is expressed through multiplication by -1. |
Ambiguity checks
When an expression can be read more than one way, COO favors rewriting rather than arguing over convention.
| Ambiguous or compressed form | Clearer rewrite |
|---|
| -\sqrt{x} | -x^{\dfrac{1}{2}} |
| \sqrt{x^2y} | (x^2y)^{\dfrac{1}{2}} |
| -x^{\dfrac{1}{2}} | -(x^{\dfrac{1}{2}}) |
| (-x)^{\dfrac{1}{2}} | Explicitly grouped signed base under a fractional exponent. |
