Math Problem Solver

Roots

\[\sqrt{2 \cdot {a}^{2}} = i \sqrt[4]{\left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} + \sqrt[4]{\left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} \approx 2.0 i \left(0.25 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} + 2.0 \left(0.25 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}\]\[\sqrt{2 \cdot {a}^{2}} = i \left(- \sqrt[4]{\left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}\right) - \sqrt[4]{\left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} \approx - 2.0 i \left(0.25 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} - 2.0 \left(0.25 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}\]