Math Problem Solver

Roots

\[\sqrt{x + \frac{3}{y} + 3 z} = i \sqrt[4]{\left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)^{2} + \left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}},\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \right)}}{2} \right)} + \sqrt[4]{\left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)^{2} + \left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}},\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \right)}}{2} \right)} \approx i \left(\left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3.0 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3.0 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3.0 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3.0 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3.0\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)^{2} + \left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3.0 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3.0 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3.0 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3.0 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3.0\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}},\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \right)}}{2} \right)} + \left(\left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3.0 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3.0 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3.0 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3.0 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3.0\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)^{2} + \left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3.0 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3.0 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3.0 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3.0 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3.0\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}},\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \right)}}{2} \right)}\]\[\sqrt{x + \frac{3}{y} + 3 z} = i \left(- \sqrt[4]{\left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)^{2} + \left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}},\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \right)}}{2} \right)}\right) - \sqrt[4]{\left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)^{2} + \left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}},\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \right)}}{2} \right)} \approx - i \left(\left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3.0 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3.0 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3.0 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3.0 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3.0\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)^{2} + \left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3.0 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3.0 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3.0 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3.0 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3.0\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}},\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \right)}}{2} \right)} - \left(\left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3.0 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3.0 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3.0 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3.0 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3.0\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)^{2} + \left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3.0 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3.0 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3.0 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3.0 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3.0\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}},\frac{\left(\operatorname{re}{\left(x\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)} + 3 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\left(- \operatorname{re}{\left(x\right)} \operatorname{re}{\left(y\right)} - 3 \operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)} + \operatorname{im}{\left(x\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)} - 3\right) \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \right)}}{2} \right)}\]