Is cube root the same as raising to power 1/3?
By Murray Bourne, 29 Oct 2015
I recently came across an interesting discrepancy involving the cube root function.
In Wolfram|Alpha (which uses the Mathematica computer algebra system at its backend), if you ask it to graph , you get the following, as expected:
In the search box, I put "cube root of x", and it stated the "Result" was correctly written as .
This graph is the reflection of the graph y = x3 in the line y = x. They are inverse functions.
We know that that cube root of a negative number is negative, so for example , and we can see this makes sense on the graph above.
Wolfram|Alpha states there is one root (x = 0), and the domain and range are all real numbers, which is consistent with the graph above.
NOTE: In tiny font, Wolfram|Alpha states:
Assuming "cube root of" is the real-valued root.
There is an option to see the "principal root", but this just gave the same result.
Raising to the power 1/3
We learn early in the study of roots and fractional powers that we can write roots in terms of fractional exponents. In general, that means:
So for square root we have:
and for cube root:
So we would expect the graph for to be the same as the one for .
But it isn't. Here's what Wolfram|Alpha returns when I ask it to graph :
The blue curve is labeled "real part" and the red curve is "imaginary part".
Curiously, the "Input" value is stated as: , but that's not actually what I entered. So part of the answer regards , but the rest of the answer does not.
We know from the section on complex roots (see especially Exercise 4 at the end), that a cubic equation will have 3 roots (just as a quadratic equation has 2 roots). These 3 roots may be all real, or a mix of real and complex roots.
Wolfram|Alpha is right to indicate there are imaginary parts, but is their graph correct? Surely the cube root of a negative number should be negative?
Example: What are all the cube roots of −8?
I zoomed out a bit to get this graph, and added some guide segments (in green):
Using the same thinking as Exercise 4 mentioned earlier, the complex solutions for x3 = −8 should be 120° apart, giving (where ):
x = −2
x = 1 + 1.73j
x = 1 − 1.73j
The above graph does give us one of these solutions (the middle one, since we can see the real part is 1 and the imaginary part is 1.73), but it doesn't give the other two solutions.
Once again, the page tells us it is assuming the "principal root", and gives us the option to choose the "real-valued root". If we do that this time, we get the real root only version, looking like the graph at the top of the page.
Scientific Notebook answer
Scientific Notebook gives the following 2 graphs, which I've overlaid.
The blue graph is , and Scientific Notebook gives the full real solution (in first and third quadrants), while the magenta (pink) graph, is in the positive quadrant only.
Geogebra and Desmos answers
Both Geogebra and Desmos give the same "full real value" graph for both and .
Similar issue to square root
I've written before about the number of solutions for √16. The answer there is one solution of course, whereas if you are asked to solve , you will get 2 solutions.
Wolfram|Alpha and Scientific Notebook recognise there is a difference between (there is one "principal" answer each time) and , where we need to remember the complex roots.
Don't just accept the computer's word for it when it gives you a graph, or the solution for some equation. It's doing its best to figure out what you want to know, but can't be expected to know the full context of your query, or necessarily give you all the possible answers.
I find it strange that Wolfram|Alpha gives only a partial graphical answer for the cube root of −8. The user has to do quite a bit of thinking to extract the complete answer from it.
See the 7 Comments below.