<s>[INST] Calculate the sum of all irrational roots of $4\sqrt[3]{8x- 3}- 8x^{3}- 3= 0$ 
Calculate the sum of all irrational roots of
$$4\sqrt[3]{8x- 3}- 8x^{3}- 3= 0$$

I'm not even sure how to begin here, I tried raising it to the power of three, tried writing $8x^{3}+ 3$ with $x^{3}+ y^{3}= \left ( x+ y \right )\left ( x^{2}- xy+ y^{2} \right ),$ but have had no meaningful results.
 [/INST] 

*

*If we assume "cube root" is the real-valued root, then we have

$$\begin{cases}4\sqrt[3]{8x-3}-8x^3-3=0 \\ x\in\mathbb R  \end{cases} \Longleftrightarrow  512 x^9 + 576 x^6 + 216 x^3 - 512 x + 219=0$$
$$\Longleftrightarrow (2 x - 1) (4 x^2 + 2 x - 3) (64 x^6 + 64 x^4 + 48 x^3 + 64 x^2 + 24 x + 73)=0$$

Then using standard calculus tools (but, only algebraic ways are also possible ) we get
$$64 x^6 + 64 x^4 + 48 x^3 + 64 x^2 + 24 x + 73 > 0$$
for all $x\in\mathbb R$.
So, we have only $2$ irrational roots:
$$x_1+x_2=\frac{-2}{4}=- \frac 12$$
Then, answer is equal to $- \dfrac 12.$



*

*If we use the principal root instead of "real-valued cube root",  then we have

$$4\sqrt[3]{8x-3}-8x^3-3=0 \Longleftrightarrow \begin{cases} 512 x^9 + 576 x^6 + 216 x^3 - 512 x + 219=0 \\ 8x-3≥0  \end{cases} $$
$$\Longleftrightarrow \begin{cases} (2 x - 1) (4 x^2 + 2 x - 3) (64 x^6 + 64 x^4 + 48 x^3 + 64 x^2 + 24 x + 73)=0 \\ x≥\frac 38\end{cases}$$

The last result implies, we have only $1$ irrational root:
$$x=\frac {-1+\sqrt {13}}{4}$$
So, answer is equal to $\dfrac {\sqrt {13}-1}{4}.$
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