$\sqrt{xˆ4-8xˆ2}$
You asked:
MathBot Answer:
Evaluated
\(\displaystyle \sqrt{{x}^{4} - 8 {x}^{2}} = \sqrt{x^{4} - 8 x^{2}} \)
Expanded
\[\sqrt{{x}^{4} - 8 {x}^{2}} = \sqrt{x^{4} - 8 x^{2}}\]
Factored
\[\sqrt{{x}^{4} - 8 {x}^{2}} = \sqrt{x^{2} \left(x^{2} - 8\right)}\]
Roots
\[\sqrt{{x}^{4} - 8 {x}^{2}} = 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} - 16 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}\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} - 8 \left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} + 8 \left(\operatorname{im}{\left(x\right)}\right)^{2}\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} - 16 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\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} - 8 \left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} + 8 \left(\operatorname{im}{\left(x\right)}\right)^{2} \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} - 16 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}\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} - 8 \left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} + 8 \left(\operatorname{im}{\left(x\right)}\right)^{2}\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} - 16 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\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} - 8 \left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} + 8 \left(\operatorname{im}{\left(x\right)}\right)^{2} \right)}}{2} \right)} \approx 4.0 i \left(\left(0.25 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 0.25 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3} - \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}\right)^{2} + 0.25 \left(0.125 \left(\operatorname{re}{\left(x\right)}\right)^{4} - 0.75 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} - \left(\operatorname{re}{\left(x\right)}\right)^{2} + 0.125 \left(\operatorname{im}{\left(x\right)}\right)^{4} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\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} - 16 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\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} - 8 \left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} + 8 \left(\operatorname{im}{\left(x\right)}\right)^{2} \right)}}{2} \right)} + 4.0 \left(\left(0.25 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 0.25 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3} - \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}\right)^{2} + 0.25 \left(0.125 \left(\operatorname{re}{\left(x\right)}\right)^{4} - 0.75 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} - \left(\operatorname{re}{\left(x\right)}\right)^{2} + 0.125 \left(\operatorname{im}{\left(x\right)}\right)^{4} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\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} - 16 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\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} - 8 \left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} + 8 \left(\operatorname{im}{\left(x\right)}\right)^{2} \right)}}{2} \right)}\]\[\sqrt{{x}^{4} - 8 {x}^{2}} = 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} - 16 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}\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} - 8 \left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} + 8 \left(\operatorname{im}{\left(x\right)}\right)^{2}\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} - 16 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\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} - 8 \left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} + 8 \left(\operatorname{im}{\left(x\right)}\right)^{2} \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} - 16 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}\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} - 8 \left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} + 8 \left(\operatorname{im}{\left(x\right)}\right)^{2}\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} - 16 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\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} - 8 \left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} + 8 \left(\operatorname{im}{\left(x\right)}\right)^{2} \right)}}{2} \right)} \approx - 4.0 i \left(\left(0.25 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 0.25 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3} - \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}\right)^{2} + 0.25 \left(0.125 \left(\operatorname{re}{\left(x\right)}\right)^{4} - 0.75 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} - \left(\operatorname{re}{\left(x\right)}\right)^{2} + 0.125 \left(\operatorname{im}{\left(x\right)}\right)^{4} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\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} - 16 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\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} - 8 \left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} + 8 \left(\operatorname{im}{\left(x\right)}\right)^{2} \right)}}{2} \right)} - 4.0 \left(\left(0.25 \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{im}{\left(x\right)} - 0.25 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{3} - \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}\right)^{2} + 0.25 \left(0.125 \left(\operatorname{re}{\left(x\right)}\right)^{4} - 0.75 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2} - \left(\operatorname{re}{\left(x\right)}\right)^{2} + 0.125 \left(\operatorname{im}{\left(x\right)}\right)^{4} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\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} - 16 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\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} - 8 \left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{4} + 8 \left(\operatorname{im}{\left(x\right)}\right)^{2} \right)}}{2} \right)}\]