Evaluate: x^3+(-3)^4=17

Expression: ${x}^{3}+{\left( -3 \right)}^{4}=17$

Evaluate the power

${x}^{3}+81=17$

Move the constant to the right-hand side and change its sign

${x}^{3}=17-81$

Calculate the difference

${x}^{3}=-64$

Take the root of both sides of the equation

$x=\sqrt[3]{-64}$

Write the complex number in polar form

$x=\sqrt[3]{64\left( \cos\left({π}\right)+i \times \sin\left({π}\right) \right)}$

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 $\begin{array} { l }k=0,& 1,& …,& n-1\end{array}$

$x=\sqrt[3]{64}\left( \cos\left({\frac{ π+2kπ }{ 3 }}\right)+i \times \sin\left({\frac{ π+2kπ }{ 3 }}\right) \right)$

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

$\begin{array} { l }x_1=\sqrt[3]{64}\left( \cos\left({\frac{ π+2 \times 0π }{ 3 }}\right)+i \times \sin\left({\frac{ π+2 \times 0π }{ 3 }}\right) \right),\\x_2=\sqrt[3]{64}\left( \cos\left({\frac{ π+2 \times 1π }{ 3 }}\right)+i \times \sin\left({\frac{ π+2 \times 1π }{ 3 }}\right) \right),\\x_3=\sqrt[3]{64}\left( \cos\left({\frac{ π+2 \times 2π }{ 3 }}\right)+i \times \sin\left({\frac{ π+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Evaluate the cube root

$\begin{array} { l }x_1=4\left( \cos\left({\frac{ π+2 \times 0π }{ 3 }}\right)+i \times \sin\left({\frac{ π+2 \times 0π }{ 3 }}\right) \right),\\x_2=\sqrt[3]{64}\left( \cos\left({\frac{ π+2 \times 1π }{ 3 }}\right)+i \times \sin\left({\frac{ π+2 \times 1π }{ 3 }}\right) \right),\\x_3=\sqrt[3]{64}\left( \cos\left({\frac{ π+2 \times 2π }{ 3 }}\right)+i \times \sin\left({\frac{ π+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Simplify the expression

$\begin{array} { l }x_1=4\left( \cos\left({\frac{ π }{ 3 }}\right)+i \times \sin\left({\frac{ π+2 \times 0π }{ 3 }}\right) \right),\\x_2=\sqrt[3]{64}\left( \cos\left({\frac{ π+2 \times 1π }{ 3 }}\right)+i \times \sin\left({\frac{ π+2 \times 1π }{ 3 }}\right) \right),\\x_3=\sqrt[3]{64}\left( \cos\left({\frac{ π+2 \times 2π }{ 3 }}\right)+i \times \sin\left({\frac{ π+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Simplify the expression

$\begin{array} { l }x_1=4\left( \cos\left({\frac{ π }{ 3 }}\right)+i \times \sin\left({\frac{ π }{ 3 }}\right) \right),\\x_2=\sqrt[3]{64}\left( \cos\left({\frac{ π+2 \times 1π }{ 3 }}\right)+i \times \sin\left({\frac{ π+2 \times 1π }{ 3 }}\right) \right),\\x_3=\sqrt[3]{64}\left( \cos\left({\frac{ π+2 \times 2π }{ 3 }}\right)+i \times \sin\left({\frac{ π+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Evaluate the cube root

$\begin{array} { l }x_1=4\left( \cos\left({\frac{ π }{ 3 }}\right)+i \times \sin\left({\frac{ π }{ 3 }}\right) \right),\\x_2=4\left( \cos\left({\frac{ π+2 \times 1π }{ 3 }}\right)+i \times \sin\left({\frac{ π+2 \times 1π }{ 3 }}\right) \right),\\x_3=\sqrt[3]{64}\left( \cos\left({\frac{ π+2 \times 2π }{ 3 }}\right)+i \times \sin\left({\frac{ π+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Simplify the expression

$\begin{array} { l }x_1=4\left( \cos\left({\frac{ π }{ 3 }}\right)+i \times \sin\left({\frac{ π }{ 3 }}\right) \right),\\x_2=4\left( \cos\left({π}\right)+i \times \sin\left({\frac{ π+2 \times 1π }{ 3 }}\right) \right),\\x_3=\sqrt[3]{64}\left( \cos\left({\frac{ π+2 \times 2π }{ 3 }}\right)+i \times \sin\left({\frac{ π+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Simplify the expression

$\begin{array} { l }x_1=4\left( \cos\left({\frac{ π }{ 3 }}\right)+i \times \sin\left({\frac{ π }{ 3 }}\right) \right),\\x_2=4\left( \cos\left({π}\right)+i \times \sin\left({π}\right) \right),\\x_3=\sqrt[3]{64}\left( \cos\left({\frac{ π+2 \times 2π }{ 3 }}\right)+i \times \sin\left({\frac{ π+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Evaluate the cube root

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

Simplify the expression

$\begin{array} { l }x_1=4\left( \cos\left({\frac{ π }{ 3 }}\right)+i \times \sin\left({\frac{ π }{ 3 }}\right) \right),\\x_2=4\left( \cos\left({π}\right)+i \times \sin\left({π}\right) \right),\\x_3=4\left( \cos\left({\frac{ 5π }{ 3 }}\right)+i \times \sin\left({\frac{ π+2 \times 2π }{ 3 }}\right) \right)\end{array}$

Simplify the expression

$\begin{align*}&\begin{array} { l }x_1=4\left( \cos\left({\frac{ π }{ 3 }}\right)+i \times \sin\left({\frac{ π }{ 3 }}\right) \right),\\x_2=4\left( \cos\left({π}\right)+i \times \sin\left({π}\right) \right),\\x_3=4\left( \cos\left({\frac{ 5π }{ 3 }}\right)+i \times \sin\left({\frac{ 5π }{ 3 }}\right) \right)\end{array} \\&\begin{array} { l }x_1=2+2\sqrt{ 3 }i,\\x_2=-4,\\x_3=2-2\sqrt{ 3 }i\end{array}\end{align*}$