2
$\begingroup$

When reducing and simplifying a complex algebraic inequality I get an expression containing

Root[-a + b #1^3 &, 1] < β < (a/b)

How can I interpret the first term in the inequality? The 1st root of some function? Why not putting it explicitly?

$\endgroup$

1 Answer 1

3
$\begingroup$

Why not putting it explicitly?

From help,

Root "Represents the exact k^th root of the polynomial equation f[x]==0"

This can be rewritten as follows

 expr = Root[-a+b#1^3&,1]< \[Beta] < (a/b)
 expr // ToRadicals

enter image description here

From help on ToRadicals it says

attempts to express all Root objects in expr in terms of radicals.

To try to answer the comment:

how the expression Root[-a+b#1^3&,1] results in the fraction shown above?

The above answer comes from, when rewriting Root[-a + b #1^3 &, 1] as

 Solve[-a + b*x^3 == 0, x]

Where #1^3 becomes x^3. Now Root[....,1] says the first root. i.e. the first root of -a + b*x^3 == 0. Which is

 Solve[-a + b*x^3 == 0, x]

enter image description here

Now the question might be, since there are 3 roots to the cubic, why the first root was choosing x -> ((-1)^(2/3) a^(1/3))/b^(1/3) and not x -> a^(1/3)/b^(1/3) ? It looks like some internal ordering is used to decide which is the first root. As Root[-a + b #1^3 &, 3] // ToRadicals gives a^(1/3)/b^(1/3)

$\endgroup$
3
  • $\begingroup$ Dear @Nasser, you actually answered my question, but may I ask you to add a couple of words on how the expression Root[-a+b#1^3&,1] results in the fraction shown above? I'm a novice in Mathematica and I have hard time trying to decode this... $\endgroup$
    – Dmitry
    Jul 16, 2020 at 10:13
  • 1
    $\begingroup$ @Dmitry I tried to answer you comment in the post. $\endgroup$
    – Nasser
    Jul 16, 2020 at 10:33
  • 1
    $\begingroup$ Ah, now I understand where this root comes from. That was the missing piece in the puzzle :) $\endgroup$
    – Dmitry
    Jul 16, 2020 at 10:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.