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Question Number 56553 by problem solverd last updated on 18/Mar/19

solve for m m^8 =3125

$$\mathrm{solve}\:\mathrm{for}\:{m} \\ $$$${m}^{\mathrm{8}} =\mathrm{3125} \\ $$

Answered by MJS last updated on 18/Mar/19

x^8 =a has got 8 solutions x_(1, 2) =±(a)^(1/8) x_(3, 4) =±(a)^(1/8) i x_(5, 6, 7, 8) =±(a)^(1/8) (((√2)/2)±((√2)/2)i) in this case m_(1, 2) =±5^(5/8) ≈±2.7344 m_(3, 4) =±5^(5/8) i≈±2.7344i m_(5, 6, 7, 8) =±5^(5/8) (((√2)/2)±((√2)/2)i)≈±1.9335±1.9335i

$${x}^{\mathrm{8}} ={a} \\ $$$$\mathrm{has}\:\mathrm{got}\:\mathrm{8}\:\mathrm{solutions} \\ $$$${x}_{\mathrm{1},\:\mathrm{2}} =\pm\sqrt[{\mathrm{8}}]{{a}} \\ $$$${x}_{\mathrm{3},\:\mathrm{4}} =\pm\sqrt[{\mathrm{8}}]{{a}}\mathrm{i} \\ $$$${x}_{\mathrm{5},\:\mathrm{6},\:\mathrm{7},\:\mathrm{8}} =\pm\sqrt[{\mathrm{8}}]{{a}}\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\pm\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\mathrm{i}\right) \\ $$$$\mathrm{in}\:\mathrm{this}\:\mathrm{case} \\ $$$${m}_{\mathrm{1},\:\mathrm{2}} =\pm\mathrm{5}^{\mathrm{5}/\mathrm{8}} \approx\pm\mathrm{2}.\mathrm{7344} \\ $$$${m}_{\mathrm{3},\:\mathrm{4}} =\pm\mathrm{5}^{\mathrm{5}/\mathrm{8}} \mathrm{i}\approx\pm\mathrm{2}.\mathrm{7344i} \\ $$$${m}_{\mathrm{5},\:\mathrm{6},\:\mathrm{7},\:\mathrm{8}} =\pm\mathrm{5}^{\mathrm{5}/\mathrm{8}} \left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\pm\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\mathrm{i}\right)\approx\pm\mathrm{1}.\mathrm{9335}\pm\mathrm{1}.\mathrm{9335i} \\ $$

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