[Maxima-bugs] [maxima:bugs] #5705 ((-1)^(1/6))^2 simplifies to -1

[David S. <tom...@us...>]

- **status**: open --> closed
- **Comment**:

Fixed by commit [d96bc5].



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**[bugs:#5705] \(\(-1\)^\(1/6\)\)^2 simplifies to -1**

**Status:** closed
**Group:** None
**Labels:** simplification simpexpt 
**Created:** Fri Oct 23, 2026 08:19 AM UTC by David Scherfgen
**Last Updated:** Fri Oct 23, 2026 08:19 AM UTC
**Owner:** nobody


By Claude:

With the default `domain : real`, Maxima reads `(-1)^(p/q)` with odd `q` as the real root, so `(-1)^(1/3)` simplifies to `-1`. `(-1)^(1/6)` has no real value and is the principal root `exp(%i*%pi/6)`, which is also how `rectform` and `float` evaluate it. But a power or a product of such roots is folded into `(-1)^(1/3)`, which is then read as the real root:

```
(%i1) ((-1)^(1/6))^2;
(%o1) -1
(%i2) (-1)^(1/6)*(-1)^(1/6);
(%o2) -1
(%i3) ((-8)^(1/6))^2;
(%o3) -2
```

The correct value of the first two is `exp(%i*%pi/3)`, that is `1/2+sqrt(3)*%i/2`:

```
(%i4) expand(float(rectform((-1)^(1/6)))^2);
(%o4) 0.8660254037844386*%i+0.4999999999999999
```

As `ratsimp` multiplies such powers out, it often returns wrong values for expressions with `(-1)^(1/6)` or `(-1)^(1/10)`. Here the difference should be `0`:

```
(%i5) r : ratsimp(1/(1+(-1)^(1/6))), algebraic : true;
(%o5) -((%i+(-1)^(5/6)+(-1)^(1/6)-1)/2)
(%i6) float(rectform(r - 1/(1+(-1)^(1/6))));
(%o6) 5.551115123125783e-17-0.8660254037844386*%i
```

With `domain : complex`, all of these are correct.


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