# fractional power to negative number

bool( (-2)^(1/3) == - 2^(1/3) ) returns False

Is there any way in sage that will say they are same?

fractional power to negative number

bool( (-2)^(1/3) == - 2^(1/3) ) returns False

Is there any way in sage that will say they are same?

add a comment

1

It depends what you mean by "same": There are 3 complex cube roots of -1, and when you input something like (-1)^(1/3), sage chooses one of them:

```
sage: CC((-1)^(1/3))
0.500000000000000 + 0.866025403784439*I
sage: CC(-1^(1/3))
-1.00000000000000
sage: var('t')
t
sage: p = (t^(3) + 1)
sage: p.roots()
[(1/2*I*(-1)^(1/3)*sqrt(3) - 1/2*(-1)^(1/3), 1), (-1/2*I*(-1)^(1/3)*sqrt(3) - 1/2*(-1)^(1/3), 1), ((-1)^(1/3), 1)]
sage: p.roots(ring=CC)
[(-1.00000000000000, 1), (0.500000000000000 - 0.866025403784439*I, 1), (0.500000000000000 + 0.866025403784439*I, 1)]
sage: p.roots(ring=RR)
[(-1.00000000000000, 1)]
```

So this is probably the reason your comparison returns `False`

:

```
sage: CC((-2)^(1/3))
0.629960524947437 + 1.09112363597172*I
sage: CC(-2^(1/3))
-1.25992104989487
```

Of course if you choose the *Real* cube root of -1, then your comparison should return `True`

. How you decide to work with this probably depends on the problem you have in mind -- can you update with some more details?

For example, could you take absolute values before taking fractional powers, and then check whether or not the signs agree?

```
sage: m = 2
sage: n = -2
sage: (abs(m))^(1/3) == (abs(n))^(1/3) and -1*sgn(m) == sgn(n)
```

Thanks, I just want to compare two numbers where they can be of the above type. And for sure I want to consider the real roots. Like CC( ), is there any function like RR( ). Seems not. Suppose I have number 1 and number2, can I use something like bool( RR(number 1) == RR(number 2) )

0

You can also try to play around with simplify_trig and simplify_radical:

```
sage: eq = (-2)^(1/3) == - 2^(1/3)
sage: bool(eq.simplify_radical())
True
sage: eq = cos(x)*(tan(x))^(1/3) == (sin(x)*(cos(x))^2)^(1/3)
sage: print eq.simplify_trig().simplify_radical()
sage: bool(eq.simplify_trig().simplify_radical())
True
```

Please start posting anonymously - your entry will be published after you log in or create a new account.

Asked: ** 2010-12-16 13:14:52 +0200 **

Seen: **1,116 times**

Last updated: **Dec 16 '10**

Why does plot choke on x to the 1/3 power, when it will calculate it just fine?

Round trip through Mathematica's FullSimplify

negative number cannot be raised to a fractional power in RDF

Power of a polynomial mod (n, X^r - 1)

Plotting an Infinite Power Tower

simplify coefficients of laurent series?

Solve an expression with fractional exponents

"Sage for power users" by W. Stein (PDF without error?)

Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.