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I believe that the cubic root of $-8$ is $-2$, ie $\sqrt[3]{-8}=-2$, or $(-2)^3=-8$.

When I ask Maple to tell me what this cubic root is

root(-8, 3)

I get the result

$$2(-1)^{1/3}$$

In Maple, let's say that we want to plot $f(x)=\sqrt[3]{x}$

plot(x^(1/3), x = -2 .. 2)

The result is

enter image description here

Why don't we see the portion of the graph where $x<0$?

Here is another example, of $g(x)=\sqrt[3]{x^2-1}$. When I plot the function in Maple

plot((x^2 - 1)^(1/3), x = -2 .. 2)

enter image description here

There is no graph between $-1$ and $1$. I must be making a silly mistake, because in my mind $g(0)=-1$.

Is there a reason for Maple not to give the expected results for cubic roots of negative numbers?

xoux
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    You might have some luck asking this question at MaplePrimes – Michael Burr Jun 01 '22 at 11:11
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    Use the function surd. – MasB Jun 01 '22 at 11:25
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    I suppose one can write cubroot function which gives $x^{1/3}$ for $x\geq 0$ and gives $-(-x)^{1/3}$ for $x<0$. – Ivan Kaznacheyeu Jun 01 '22 at 11:30
  • Maybe you forget that $(-1)^{1/3}=e^{i\pi/3}$. I.e. you get complex numbers – drC1Ron Jun 01 '22 at 11:36
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    About reason: $z^{1/3}$ is multi-valued function, maybe default setting of Maple is to give complex root for negative $z$. If you calculate $z^{1/3}$ as $|z|^{1/3}\exp (i \frac13 \arg z)$ and use principal value of $\arg z$ for negative real number $z=x<0$, then first root is $|x|^{1/3}\exp (i\frac{\pi}3)$ which is complex number. – Ivan Kaznacheyeu Jun 01 '22 at 11:36
  • @RonnyLandsverk But you also get complex values for $x^{1/3}$, $x>0$. – David C. Ullrich Jun 01 '22 at 11:48
  • @DavidC.Ullrich Do you really ? For real $x>0$ ? – drC1Ron Jun 01 '22 at 12:02
  • The function surd works like a charm. – xoux Jun 01 '22 at 12:13
  • Yes, For example $1^{1/3}$ has the three values $1$, $e^{\pm2\pi i/3}$. Of course usually we just consider the real cube root, but again that doesn't show that $x<0$ is different from $x>0$; for any real $x$ we usually just consider the real cube root. – David C. Ullrich Jun 01 '22 at 12:36
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    You specifically mention the call root(-8,3). The Help page for the root command explicitly describes this situation and this very same example. It describes that root returns the principal root, with formula. It suggests using surd to get the real root here. https://www.maplesoft.com/support/help/maple/view.aspx?path=root#info – acer Jun 01 '22 at 19:51

1 Answers1

1

See the Maple function surd.

plot(surd(x,3),x=-2..2);

A

plot(surd(x^2 - 1,3), x = -2 .. 2);

B

GEdgar
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