Intereting Posts

$a-b,a^2-b^2,a^3-b^3…$ are integers $\implies$ $a,b$ are integers?
Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?
Describe the Riemann surface for $w^2=z^2-1$.
Prove every group of order less or equal to five is abelian
Legendre symbol, second supplementary law
Can a set containing $0$ be purely imaginary?
Lambert W function with rational polynomial
Show that, given spherically symmetric initial data, a solution to the heat equation is spherically symmetric
Testing whether a hypersurface is singular
How to apply reduction of order to find a 2nd linearly independent solution?
Cardinality of power set of $\mathbb N$ is equal to cardinality of $\mathbb R$
find the fourier cosine transform of the function defined by $\displaystyle f(x)= \frac1{1+x^2}$
Sum of $\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$
$\sum_{i=1}^n \frac{n}{\text{gcd}(i,n)}.$
Induction without a base case

In my precalc book, I have the following problem:

Calculate $a+b+c$ if $a,b,c\in\mathbb{Q}$ and

$$\sqrt[3]{\sqrt[3]{2}-1}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$$

I think that the RHS can stay untouched, while operating the LHS, but I can’t find a way to factor $\sqrt[3]{2}-1$ as the third power of something. Any help is greatly appreciated.

- Calculate $\tan9^{\circ}-\tan27^{\circ}-\tan63^{\circ}+\tan81^{\circ}$
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- Collatz-ish Olympiad Problem
- $ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ$
- Is there any explicit formula for $x_n$?

With the help of Olegg, i got the solution

$$\sqrt[3]{\sqrt[3]{2}-1}=\sqrt[3]{\frac{(\sqrt[3]{2}-1)(\sqrt[3]{4}+\sqrt[3]{2}+1)}{\sqrt[3]{4}+\sqrt[3]{2}+1}}$$

$$\sqrt[3]{\frac{1}{\sqrt[3]{4}+\sqrt[3]{2}+1}}$$

$$\sqrt[3]{\frac{1}{(\sqrt[3]{\frac{1}{3}}+\sqrt[3]{\frac{2}{3}})^3}}$$

$$\frac{1}{\sqrt[3]{\frac{1}{3}}+\sqrt[3]{\frac{2}{3}}}$$

$${\sqrt[3]{\frac{1}{9}}-\sqrt[3]{\frac{2}{9}}+\sqrt[3]{\frac{4}{9}}}$$

$$a+b+c=\frac{1}{3}$$

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- Determine where a point lies in relation to a circle, is my answer right?
- Solving Equation through inequalities.
- What are the rules for factorial manipulation?
- If $a\in R$ and the equation $-3(x-\lfloor x \rfloor)^2+2(x-\lfloor x \rfloor)+a^2=0$ has no integral solution,then all possible values of $a$
- A polynomial determined by two values
- Show that there exist only $n$ solutions
- How can I find the following product? $ \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 80^\circ.$
- If 5 points are necessary to determine a conic, why are only 3 necessary to determine a parabola?
- Differentiation using first principles with rational powers

**Hint.**

$(a,b,c) = \Bigl(\dfrac{1}{9},-\dfrac{2}{9},\dfrac{4}{9}\Bigr)$ $-$ one of rational solutions (ignoring permutations).

So, $a+b+c=\dfrac{1}{3}$.

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- Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers
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