CounterexampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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$((-1)^2)^{1/2}=1\ne-1=(-1)^{2(1/2)}$ for principal complex powers

Statement refuted

The principal-power law $(z^a)^b=z^{ab}$ is false without branch hypotheses. The conventions and prerequisite facts used below are recorded in Complex logarithms, the principal logarithm, and principal and multivalued complex powers, All logarithms of $z\ne0$ are $\operatorname{Log}z+2\pi i k$ , $k\in\mathbb Z$ .

Facts & Assumptions

Given: $z=-1$ , $a=2$ , and $b=1/2$ .

Counterexample

1.1

The principal square of $-1$ is $1$ , so $((-1)^2)^{1/2}=1$ .

given

2.1

The principal value of $(-1)^1$ is $-1$ , so the two sides differ.

given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 23 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)

((−1)2)1/2=1≠−1=(−1)2(1/2)((-1)^2)^{1/2}=1\ne-1=(-1)^{2(1/2)}((−1)2)1/2=1=−1=(−1)2(1/2) for principal complex powers