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Plot of x^(1/3)

Yankel

Active member
Jan 27, 2012
398
Dear all,

I was using the computer in order to plot the graph of

\[y=x^{\frac{1}{3}}=\sqrt[3]{x}\]

and two different plotters gave two different results. I don't understand why. Can you kindly explain ?

The results are:

plot 1.PNG

plot 2.PNG

Thank you !
 

Country Boy

Well-known member
MHB Math Helper
Jan 30, 2018
368
First, of course, those are the same for x> 0 just scaled differently. As for x< 0 it looks like the first plotter is using logarithms "unthinkingly" to calculate fractional powers and the logarithm of negative numbers do not exist.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,687
Generally $x^{1/3}$ is considered to be undefined for negative $x$, although some books do effectively define it as $-(-x)^{1/3}$.

Reasons are:
  1. The power identity $a^{b\cdot c}=(a^b)^c$ breaks down. Consider:
    $$-1 = (-1)^{2/3\cdot 3/2} \ne ((-1)^{2/3})^{3/2} = 1^{3/2} = 1$$
  2. Calculators evaluate it as $x^{0.33333}$, which is undefined for negative x.
    Note that we can only define something like $x^{1/3}$ for negative x if the power is a fraction with an odd number in the denominator, but that is generally not supported by calculators.