An Enigmatic radical expression.

Let $x$ be the number we need to find and $(a+b\sqrt{12345679})^2 = 12345689654321233 + 5333334096 \sqrt{12345679}$ . Then

$\begin{aligned} x & = \sqrt[3]{\sqrt{a+b\sqrt{12345679}} - \sqrt{|a-b\sqrt{12345679}|}} & \small \color{#3D99F6} \text{Let } (c+d\sqrt{12345679})^2 = a+b\sqrt{12345679} \\ & = \sqrt[3]{c+d\sqrt{12345679} - \left|c-d\sqrt{12345679}\right|} \end{aligned}$

$\begin{aligned} (a+b\sqrt{12345679})^2 & = 12345689654321233 + 5333334096 \sqrt{12345679} \\ a^2+ 12345679 b^2 + 2ab\sqrt{12345679} & = 12345689654321233 + 5333334096 \sqrt{12345679} \end{aligned}$

$\implies \begin{cases} a^2+ 12345679 b^2 = 12345689654321233 & ...(1) \\ ab = 2666667048 & ...(2) \end{cases}$

$\begin{aligned} a^2(1): a^4+ 12345679 ({\color{#3D99F6}ab})^2 & = 12345689654321233a^2 & \small \color{#3D99F6} (2): \ ab = 2666667048 \\ a^4 - 12345689654321233a^2 + 87791520219480508137172416 & = 0 \\ \implies a & = 111111127 \\ b & = 12 \end{aligned}$

Similarly,

$\begin{aligned} (c+d\sqrt{12345679})^2 & = 111111127 + 24 \sqrt{12345679} \\ c^2+ 12345679 d^2 + 2cd\sqrt{12345679} & = 111111127 + 24 \sqrt{12345679} \end{aligned}$

$\implies \begin{cases} c^2+ 12345679 d^2 = 111111127 & ...(3) \\ cd = 24 & ...(4) \end{cases}$

$\begin{aligned} c^2(1): c^4+ 12345679 ({\color{#3D99F6}cd})^2 & = 111111127c^2 & \small \color{#3D99F6} (2): \ ab = 2666667048 \\ c^4 - 111111127c^2 + 1777777776 & = 0 \\ \implies c & = 4 \\ d & = 3 \end{aligned}$

Therefore

$\begin{aligned} x & = \sqrt[3]{4+3\sqrt{12345679} - \left|4-3\sqrt{12345679}\right|} \\ & = \sqrt[3]{4+3\sqrt{12345679} - (3\sqrt{12345679}-4)} \\ & = \sqrt[3] 8 = \boxed 2 \end{aligned}$