Simplifying roots 3

$\Rightarrow\sqrt[3]{8+\dfrac{72\sqrt3+585}{64\sqrt3}}=\sqrt[3]{8+\dfrac98+\dfrac{585}{64\sqrt3}}=\sqrt[3]{\dfrac{73}8+\dfrac{585}{64\sqrt3}}$

$\Rightarrow\sqrt[3]{8+\dfrac{72\sqrt3-585}{64\sqrt3}}=\sqrt[3]{8+\dfrac98-\dfrac{585}{64\sqrt3}}=\sqrt[3]{\dfrac{73}8-\dfrac{585}{64\sqrt3}}$

Let $\color{#D61F06}{A=\sqrt[3]{\dfrac{73}8+\dfrac{585}{64\sqrt3}}}$ , $\color{#3D99F6}{B=\sqrt[3]{\dfrac{73}8-\dfrac{585}{64\sqrt3}}}$ and $\color{#20A900}S=\color{#D61F06}A+\color{#3D99F6}B=\color{#D61F06}{\sqrt[3]{\dfrac{73}8+\dfrac{585}{64\sqrt3}}}+\color{#3D99F6}{\sqrt[3]{\dfrac{73}8-\dfrac{585}{64\sqrt3}}}$

Now,

$\color{#20A900}S^3=A^3+B^3+3AB(\color{#D61F06}A+\color{#3D99F6}B)=\color{#302B94}{A^3+B^3}+\color{magenta}{3AB}\color{#20A900}S\Rightarrow(*)$

Thus,

$\color{#302B94}{A^3+B^3}=\left(\color{#D61F06}{\sqrt[3]{\dfrac{73}8+\dfrac{585}{64\sqrt3}}}\right)^3+\left(\color{#3D99F6}{\sqrt[3]{\dfrac{73}8-\dfrac{585}{64\sqrt3}}}\right)^3=\color{#302B94}{\dfrac{73}4}$

$\begin{aligned}\color{magenta}{3AB}=&3\left(\color{#D61F06}{\sqrt[3]{\dfrac{73}8+\dfrac{585}{64\sqrt3}}}\right)\left(\color{#3D99F6}{\sqrt[3]{\dfrac{73}8-\dfrac{585}{64\sqrt3}}}\right)=3\sqrt[3]{\left(\dfrac{73}8\right)^2-\left(\dfrac{585}{64\sqrt3}\right)^2}\\=&3\sqrt[3]{\dfrac{5329}{64}-\dfrac{114075}{4096}}=3\sqrt[3]{\dfrac{226981}{4096}}=3\left(\dfrac{61}{16}\right)=\color{magenta}{\dfrac{183}{16}}\end{aligned}$

Instead $\color{#302B94}{A^3+B^3}$ with $\color{#302B94}{\dfrac{73}4}$ and $\color{magenta}{3AB}$ with $\color{magenta}{\dfrac{183}{16}}$ in equation $(*)$

$\begin{aligned}\color{#20A900}S^3=&\ \color{#302B94}{\dfrac{73}4}+\left(\color{magenta}{\dfrac{183}{16}}\right)\color{#20A900}S\\16\color{#20A900}S^3=&\ 292+183\color{#20A900}S\\16\color{#20A900}S^3-183\color{#20A900}S-292=&\ 0\\(\color{#20A900}S-4)(16\color{#20A900}S^2+64\color{#20A900}S+73)=&\ 0\end{aligned}$

From equation

$\Rightarrow\color{#20A900}S-4=0\Rightarrow\color{#20A900}{S=4}$

$\Rightarrow16\color{#20A900}S^2+64\color{#20A900}S+73=0\Rightarrow\Delta=64^2-4(16)(73)=-576<0\quad\text{[Complex Solution]}$

Hence, $\sqrt[3]{8+\dfrac{72\sqrt3+585}{64\sqrt3}}+\sqrt[3]{8+\dfrac{72\sqrt3-585}{64\sqrt3}}=\boxed4$

i don't have much time to do these types of math