$\sqrt{x^(2m-4)}$
You asked:
MathBot Answer:
Evaluated
\(\displaystyle \sqrt{{x}^{2 m - 4}} = \sqrt{x^{2 m - 4}} \)
Expanded
\[\sqrt{{x}^{2 m - 4}} = \sqrt{\frac{x^{2 m}}{x^{4}}}\]
Factored
\[\sqrt{{x}^{2 m - 4}} = \sqrt{\frac{x^{2 m}}{x^{4}}}\]
Roots
\[\sqrt{{x}^{2 m - 4}} = i \sqrt[4]{\left(\operatorname{re}{\left(\frac{x^{2 m}}{x^{4}}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{x^{2 m}}{x^{4}}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{x^{2 m}}{x^{4}}\right)},\operatorname{re}{\left(\frac{x^{2 m}}{x^{4}}\right)} \right)}}{2} \right)} + \sqrt[4]{\left(\operatorname{re}{\left(\frac{x^{2 m}}{x^{4}}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{x^{2 m}}{x^{4}}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{x^{2 m}}{x^{4}}\right)},\operatorname{re}{\left(\frac{x^{2 m}}{x^{4}}\right)} \right)}}{2} \right)} \approx i \left(\left(\operatorname{re}{\left(\frac{x^{2 m}}{x^{4}}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{x^{2 m}}{x^{4}}\right)}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{x^{2 m}}{x^{4}}\right)},\operatorname{re}{\left(\frac{x^{2 m}}{x^{4}}\right)} \right)}}{2} \right)} + \left(\left(\operatorname{re}{\left(\frac{x^{2 m}}{x^{4}}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{x^{2 m}}{x^{4}}\right)}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{x^{2 m}}{x^{4}}\right)},\operatorname{re}{\left(\frac{x^{2 m}}{x^{4}}\right)} \right)}}{2} \right)}\]\[\sqrt{{x}^{2 m - 4}} = i \left(- \sqrt[4]{\left(\operatorname{re}{\left(\frac{x^{2 m}}{x^{4}}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{x^{2 m}}{x^{4}}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{x^{2 m}}{x^{4}}\right)},\operatorname{re}{\left(\frac{x^{2 m}}{x^{4}}\right)} \right)}}{2} \right)}\right) - \sqrt[4]{\left(\operatorname{re}{\left(\frac{x^{2 m}}{x^{4}}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{x^{2 m}}{x^{4}}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{x^{2 m}}{x^{4}}\right)},\operatorname{re}{\left(\frac{x^{2 m}}{x^{4}}\right)} \right)}}{2} \right)} \approx - i \left(\left(\operatorname{re}{\left(\frac{x^{2 m}}{x^{4}}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{x^{2 m}}{x^{4}}\right)}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{x^{2 m}}{x^{4}}\right)},\operatorname{re}{\left(\frac{x^{2 m}}{x^{4}}\right)} \right)}}{2} \right)} - \left(\left(\operatorname{re}{\left(\frac{x^{2 m}}{x^{4}}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{x^{2 m}}{x^{4}}\right)}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{x^{2 m}}{x^{4}}\right)},\operatorname{re}{\left(\frac{x^{2 m}}{x^{4}}\right)} \right)}}{2} \right)}\]