Math Problem Solver

Roots

\[\sqrt{{x}^{4}} = i \sqrt[4]{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3},\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} \right)}}{2} \right)} + \sqrt[4]{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3},\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} \right)}}{2} \right)} \approx 2.44948974278318 i \left(0.444444444444444 \left(\left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(0.166666666666667 \left(\operatorname{re}{\left(x\right)}\right)^{4} - \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + 0.166666666666667 \left(\operatorname{im}{\left(x\right)}\right)^{4}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3},\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} \right)}}{2} \right)} + 2.44948974278318 \left(0.444444444444444 \left(\left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(0.166666666666667 \left(\operatorname{re}{\left(x\right)}\right)^{4} - \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + 0.166666666666667 \left(\operatorname{im}{\left(x\right)}\right)^{4}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3},\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} \right)}}{2} \right)}\]\[\sqrt{{x}^{4}} = i \left(- \sqrt[4]{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3},\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} \right)}}{2} \right)}\right) - \sqrt[4]{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3},\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} \right)}}{2} \right)} \approx - 2.44948974278318 i \left(0.444444444444444 \left(\left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(0.166666666666667 \left(\operatorname{re}{\left(x\right)}\right)^{4} - \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + 0.166666666666667 \left(\operatorname{im}{\left(x\right)}\right)^{4}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3},\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} \right)}}{2} \right)} - 2.44948974278318 \left(0.444444444444444 \left(\left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(0.166666666666667 \left(\operatorname{re}{\left(x\right)}\right)^{4} - \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + 0.166666666666667 \left(\operatorname{im}{\left(x\right)}\right)^{4}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 4 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3},\left(\operatorname{re}{\left(x\right)}\right)^{4} - 6 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} \right)}}{2} \right)}\]