All Questions
41
questions
2
votes
2
answers
100
views
Convert an expression with radicals into simpler form
It was pointed out in a mathologer video on the cubic formula that $\sqrt[3]{20 + \sqrt{392}} + \sqrt[3]{20 - \sqrt{392}}$ is actually equal to $4$. Is there a series of transformations that can be ...
- 23
2
votes
1
answer
104
views
Are there any ways to convert inverse trigonometric values to radicals?
When we solve a cubic equation $ax^3+bx^2+cx+d=0$, the roots are supposed to be in the form of radicals in real numbers or complex realm. However, if the discriminant is less than 0, the solution is ...
- 129
10
votes
0
answers
231
views
On the solvable octic $x^8-44x-33 = 0$ and the tribonacci constant
I had discussed the solvable octic trinomial,
$$x^8-44x-33=0\tag1$$
way back in this old MSE post, but I revisited this inspired by another solvable octic,
$$y^8-y^7+29y^2+29=0\tag2$$
which I also ...
- 54.9k
5
votes
1
answer
127
views
Find the value of: $\sqrt[3]{a+b}+\sqrt[3]{b+c}+\sqrt[3]{a+c}$
Let $a,b,c$ be roots of the cubic
$$x^3-x^2-2x+1=0$$
Then, find the value of:
$$\sqrt[3]{a+b}+\sqrt[3]{b+c}+\sqrt[3]{a+c}$$
My attempt.
I used the substitutions $$a+b=x^3, b+c=y^3, a+c=z^3$$
$$x^3+y^...
- 652
2
votes
3
answers
105
views
Is $a+5^{1/3}b+5^{2/3}c$ a root of any cubic polynomial in $\mathbb{Q}$?
For arbitrary $a, b, c \in \mathbb{Q}$, let $w := a + 5^{1/3} b + 5^{2/3} c$, is $w$ a root of any cubic polynomial in $\mathbb{Q}$?
I guess the cubic polynomial always exists. But I am confused about ...
- 515
0
votes
1
answer
73
views
A perplexity over the cubic root.
Good morning; I'm having doubts about a perhaps simple question involving the cubic root of a function. Say I do have the following
$$f(x) = \sqrt[3]{\dfrac{x^3}{2-x^2}}$$
Its domain is clearly $x\neq\...
- 3,482
4
votes
2
answers
492
views
Understanding Cardano's Formula
In deriving his formula, Cardano arrives at the equation $y^3+py+q=0$. By substituting $y=\sqrt[3]{u}+\sqrt[3]{v}$, he gets the equation $(u+v+q)+(\sqrt[3]{u}\sqrt[3]{v})(3\sqrt[3]{u} \sqrt[3]{v} +p)=...
- 1,467
0
votes
0
answers
55
views
Expressing the solution to a rational expression in radical form
I'm trying to find the solution to this equation
$$
-\frac{3}{r} + \frac{8}{r^3} = \frac{\sqrt{2}-1}{2}
$$
but I haven't been able to find a solution in radical form. Although I've found the solutions ...
- 1
5
votes
3
answers
662
views
Generalized denesting formula for $\sqrt[3]{A+B\sqrt{C}}$
Question: What is the generalized formula for denesting
$$\sqrt[3]{A+B\sqrt{C}}$$
Recently I've posted a question about nested radicals in solving the cubic equation. I received a comment about an ...
- 741
5
votes
0
answers
271
views
$\sqrt[3]{\text{something}\pm\sqrt{\text{something}}}$
$$\sqrt{A\pm\sqrt{B}}=\sqrt{\frac{A+C}{2}}\pm\sqrt{\frac{A-C}{2}}$$
, where
$C^2=A^2-B$.
But, I couldn't find a formula for
$\sqrt[3]{A\pm\sqrt{B}}$.
First of all, why would you even need this? I've ...
- 741
0
votes
1
answer
65
views
How to get rid of such radicals?
I would like to know if there is any way I can get rid of these cubic radicals bellow (1). I am allowing both complex and real values.
$$ \sqrt[3]{ -\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27} }...
- 516
2
votes
4
answers
1k
views
Multiple definitions of casus irreducibilis
In the case of cubic equations,
Casus irreducibilis occurs when none of the roots is rational and when all three roots are distinct and real (...)
—Wikipedia's Casus irreducibilis article
So, $x^...
- 317
-1
votes
1
answer
67
views
Find $m^3$ if $m=\sqrt[3]{a + \frac{a+8}{3}\sqrt{\frac{a-1}{3}}} + \sqrt[3]{a - \frac{a+8}{3}\sqrt\frac{a-1}{3}}$
Please help me solve this question in a easy way:
$$
\sqrt[3]{a + \frac{a+8}{3}\sqrt{\frac{a-1}{3}}} + \sqrt[3]{a - \frac{a+8}{3}\sqrt\frac{a-1}{3}} = m
$$
Find $m^3$.
(The answer is $8$.)
I ...
- 19
18
votes
4
answers
612
views
Why should it be $\sqrt[3]{6+x}=x$?
Find all the real solutions to:
$$x^3-\sqrt[3]{6+\sqrt[3]{x+6}}=6$$
Can you confirm the following solution? I don't understand line 3. Why should it be $\sqrt[3]{6+x}=x$?
Thank you.
$$
\begin{...
user548054
1
vote
2
answers
68
views
Solve$(10+6\sqrt3)^{\frac{1}{3}}-(-10+6\sqrt3)^{\frac{1}{3}}$
We need to solve the following equation $y=(10+6\sqrt3)^{\frac{1}{3}}-(-10+6\sqrt3)^{\frac{1}{3}}$ and it is equal to 2 while I am getting the value in excel I am not able to solve it manually ...
- 8,912