Calculate: 414 Request-URI Too Large

Expression: $x \times 2x \times 3x=2$

Divide both sides of the equation by $2 \times 3$

${x}^{3}=\frac{ 1 }{ 3 }$

Take the root of both sides of the equation

$x=\sqrt[3]{\frac{ 1 }{ 3 }}$

Write the complex number in polar form

$x=\sqrt[3]{\frac{ 1 }{ 3 }\left( \cos\left({0}\right)+i \times \sin\left({0}\right) \right)}$

Calculate the $n$th roots of a complex number $r\left( \cos\left({θ}\right)+i \times \sin\left({θ}\right) \right)$, using $\sqrt[n]{z}=\sqrt[n]{r}\left( \cos\left({\frac{ θ+2kπ }{ n }}\right)+i \times \sin\left({\frac{ θ+2kπ }{ n }}\right) \right)$, where $\begin{array} { l }k=0,& 1,& …,& n-1\end{array}$

$x=\sqrt[3]{\frac{ 1 }{ 3 }}\left( \cos\left({\frac{ 0+2kπ }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2kπ }{ 3 }}\right) \right)$

Since $n=3$, substitute $k=\begin{array} { l }0,& 1,& 2\end{array}$ into the expression

$\begin{array} { l }x_1=\sqrt[3]{\frac{ 1 }{ 3 }}\left( \cos\left({\frac{ 0+2 \times 0π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 0π }{ 3 }}\right) \right),\\x_2=\sqrt[3]{\frac{ 1 }{ 3 }}\left( \cos\left({\frac{ 0+2 \times 1π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 1π }{ 3 }}\right) \right),\\x_3=\sqrt[3]{\frac{ 1 }{ 3 }}\left( \cos\left({\frac{ 0+2 \times 2π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Simplify the expression

$\begin{array} { l }x_1=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({\frac{ 0+2 \times 0π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 0π }{ 3 }}\right) \right),\\x_2=\sqrt[3]{\frac{ 1 }{ 3 }}\left( \cos\left({\frac{ 0+2 \times 1π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 1π }{ 3 }}\right) \right),\\x_3=\sqrt[3]{\frac{ 1 }{ 3 }}\left( \cos\left({\frac{ 0+2 \times 2π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Simplify the expression

$\begin{array} { l }x_1=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({0}\right)+i \times \sin\left({\frac{ 0+2 \times 0π }{ 3 }}\right) \right),\\x_2=\sqrt[3]{\frac{ 1 }{ 3 }}\left( \cos\left({\frac{ 0+2 \times 1π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 1π }{ 3 }}\right) \right),\\x_3=\sqrt[3]{\frac{ 1 }{ 3 }}\left( \cos\left({\frac{ 0+2 \times 2π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Simplify the expression

$\begin{array} { l }x_1=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({0}\right)+i \times \sin\left({0}\right) \right),\\x_2=\sqrt[3]{\frac{ 1 }{ 3 }}\left( \cos\left({\frac{ 0+2 \times 1π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 1π }{ 3 }}\right) \right),\\x_3=\sqrt[3]{\frac{ 1 }{ 3 }}\left( \cos\left({\frac{ 0+2 \times 2π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Simplify the expression

$\begin{array} { l }x_1=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({0}\right)+i \times \sin\left({0}\right) \right),\\x_2=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({\frac{ 0+2 \times 1π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 1π }{ 3 }}\right) \right),\\x_3=\sqrt[3]{\frac{ 1 }{ 3 }}\left( \cos\left({\frac{ 0+2 \times 2π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Simplify the expression

$\begin{array} { l }x_1=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({0}\right)+i \times \sin\left({0}\right) \right),\\x_2=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({\frac{ 2π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 1π }{ 3 }}\right) \right),\\x_3=\sqrt[3]{\frac{ 1 }{ 3 }}\left( \cos\left({\frac{ 0+2 \times 2π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Simplify the expression

$\begin{array} { l }x_1=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({0}\right)+i \times \sin\left({0}\right) \right),\\x_2=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({\frac{ 2π }{ 3 }}\right)+i \times \sin\left({\frac{ 2π }{ 3 }}\right) \right),\\x_3=\sqrt[3]{\frac{ 1 }{ 3 }}\left( \cos\left({\frac{ 0+2 \times 2π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Simplify the expression

$\begin{array} { l }x_1=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({0}\right)+i \times \sin\left({0}\right) \right),\\x_2=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({\frac{ 2π }{ 3 }}\right)+i \times \sin\left({\frac{ 2π }{ 3 }}\right) \right),\\x_3=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({\frac{ 0+2 \times 2π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Simplify the expression

$\begin{array} { l }x_1=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({0}\right)+i \times \sin\left({0}\right) \right),\\x_2=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({\frac{ 2π }{ 3 }}\right)+i \times \sin\left({\frac{ 2π }{ 3 }}\right) \right),\\x_3=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({\frac{ 4π }{ 3 }}\right)+i \times \sin\left({\frac{ 0+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Simplify the expression

$\begin{align*}&\begin{array} { l }x_1=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({0}\right)+i \times \sin\left({0}\right) \right),\\x_2=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({\frac{ 2π }{ 3 }}\right)+i \times \sin\left({\frac{ 2π }{ 3 }}\right) \right),\\x_3=\frac{ \sqrt[3]{9} }{ 3 }\left( \cos\left({\frac{ 4π }{ 3 }}\right)+i \times \sin\left({\frac{ 4π }{ 3 }}\right) \right)\end{array} \\&\begin{array} { l }x_1=\frac{ \sqrt[3]{9} }{ 3 },\\x_2=-\frac{ \sqrt[3]{9} }{ 6 }+\frac{ \sqrt[3]{9}\sqrt{ 3 } }{ 6 } \times i,\\x_3=-\frac{ \sqrt[3]{9} }{ 6 }-\frac{ \sqrt[3]{9}\sqrt{ 3 } }{ 6 } \times i\end{array}\end{align*}$