Math Problem Solver

Roots

\[\sqrt{\frac{1}{{x}^{2}}} = i \sqrt[4]{\frac{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},\frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)} + \sqrt[4]{\frac{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},\frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)} \approx 0.707106781186548 i \left(\frac{0.25 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}{\left(0.25 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(0.25 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},\frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)} + 0.707106781186548 \left(\frac{0.25 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}{\left(0.25 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(0.25 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},\frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)}\]\[\sqrt{\frac{1}{{x}^{2}}} = i \left(- \sqrt[4]{\frac{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},\frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)}\right) - \sqrt[4]{\frac{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},\frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)} \approx - 0.707106781186548 i \left(\frac{0.25 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}{\left(0.25 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(0.25 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},\frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)} - 0.707106781186548 \left(\frac{0.25 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}{\left(0.25 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(0.25 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},\frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)}\]