Nesting Radicals

$\color{#20A900}{x}= \sqrt[3]{4 + 2\color{#20A900}{\sqrt[3]{4 + 2\sqrt[3]{4 +2\sqrt[3]{4 +2\sqrt[3]{4 + \cdots }}}}}}$

$\color{#20A900}{x}=\sqrt[3]{4+2\color{#20A900}{x}}$

Cubing both sides.

$\color{#20A900}{x}^3=4+2\color{#20A900}{x}$

$\color{#20A900}{x}^3-2\color{#20A900}{x}-4=0$

$(\color{#20A900}{x}-2)(\color{#20A900}{x}^2+2\color{#20A900}{x}+2)=0$

$\therefore \color{#20A900}{x}=2$

$\Rightarrow \sqrt[3]{4 + 2\sqrt[3]{4 + 2\sqrt[3]{4 +2\sqrt[3]{4 +2\sqrt[3]{4 + \cdots }}}}}=\color{#20A900}{2}$

$x=\sqrt[3]{4+2x}$

$x^{3}=4+2x$

$x^{3}-2x-4=0$

$(x-2)(x^{2}+2x+2)=0$

$x^{2}+2x+2$ has no solutions (as $b^{2}-4ac=2^{2}-4(2)=-4$ ), so $x=2$ is the only solution.

Note that this is actually an iterative formula, which provides a numerical way of solving an equation. Type in a random number on your calculator, and use the "answer" function to substitute it into $\sqrt[3]{4+2(ans)}$ . Then repeatedly press $=$ , and the value on the calculator display will converge on 2.