$\sqrt{25x^{10n}y^{8m}}$

asked by guest
on Nov 17, 2024 at 8:14 am



You asked:

Evaluate the expression: \(\sqrt{25 \cdot {x}^{10 n} \cdot {y}^{8 m}}\)

MathBot Answer:

Evaluated



\(\displaystyle \sqrt{25 \cdot {x}^{10 n} \cdot {y}^{8 m}} = 5 \sqrt{x^{10 n} y^{8 m}} \)

Roots

\[\sqrt{25 \cdot {x}^{10 n} \cdot {y}^{8 m}} = i \sqrt[4]{625 \left(\operatorname{re}{\left(x^{10 n} y^{8 m}\right)}\right)^{2} + 625 \left(\operatorname{im}{\left(x^{10 n} y^{8 m}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(25 \operatorname{im}{\left(x^{10 n} y^{8 m}\right)},25 \operatorname{re}{\left(x^{10 n} y^{8 m}\right)} \right)}}{2} \right)} + \sqrt[4]{625 \left(\operatorname{re}{\left(x^{10 n} y^{8 m}\right)}\right)^{2} + 625 \left(\operatorname{im}{\left(x^{10 n} y^{8 m}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(25 \operatorname{im}{\left(x^{10 n} y^{8 m}\right)},25 \operatorname{re}{\left(x^{10 n} y^{8 m}\right)} \right)}}{2} \right)} \approx 5.0 i \left(\left(\operatorname{re}{\left(x^{10 n} y^{8 m}\right)}\right)^{2} + \left(\operatorname{im}{\left(x^{10 n} y^{8 m}\right)}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(25 \operatorname{im}{\left(x^{10 n} y^{8 m}\right)},25 \operatorname{re}{\left(x^{10 n} y^{8 m}\right)} \right)}}{2} \right)} + 5.0 \left(\left(\operatorname{re}{\left(x^{10 n} y^{8 m}\right)}\right)^{2} + \left(\operatorname{im}{\left(x^{10 n} y^{8 m}\right)}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(25 \operatorname{im}{\left(x^{10 n} y^{8 m}\right)},25 \operatorname{re}{\left(x^{10 n} y^{8 m}\right)} \right)}}{2} \right)}\]\[\sqrt{25 \cdot {x}^{10 n} \cdot {y}^{8 m}} = i \left(- \sqrt[4]{625 \left(\operatorname{re}{\left(x^{10 n} y^{8 m}\right)}\right)^{2} + 625 \left(\operatorname{im}{\left(x^{10 n} y^{8 m}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(25 \operatorname{im}{\left(x^{10 n} y^{8 m}\right)},25 \operatorname{re}{\left(x^{10 n} y^{8 m}\right)} \right)}}{2} \right)}\right) - \sqrt[4]{625 \left(\operatorname{re}{\left(x^{10 n} y^{8 m}\right)}\right)^{2} + 625 \left(\operatorname{im}{\left(x^{10 n} y^{8 m}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(25 \operatorname{im}{\left(x^{10 n} y^{8 m}\right)},25 \operatorname{re}{\left(x^{10 n} y^{8 m}\right)} \right)}}{2} \right)} \approx - 5.0 i \left(\left(\operatorname{re}{\left(x^{10 n} y^{8 m}\right)}\right)^{2} + \left(\operatorname{im}{\left(x^{10 n} y^{8 m}\right)}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(25 \operatorname{im}{\left(x^{10 n} y^{8 m}\right)},25 \operatorname{re}{\left(x^{10 n} y^{8 m}\right)} \right)}}{2} \right)} - 5.0 \left(\left(\operatorname{re}{\left(x^{10 n} y^{8 m}\right)}\right)^{2} + \left(\operatorname{im}{\left(x^{10 n} y^{8 m}\right)}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(25 \operatorname{im}{\left(x^{10 n} y^{8 m}\right)},25 \operatorname{re}{\left(x^{10 n} y^{8 m}\right)} \right)}}{2} \right)}\]

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