Question Number 168399 by Mastermind last updated on 09/Apr/22
$${solve} \\ $$$${z}^{\frac{\mathrm{1}}{\mathrm{4}}} −{i}=\mathrm{0} \\ $$$$ \\ $$$${Mastermind} \\ $$
Answered by nurtani last updated on 10/Apr/22
$${i}=\sqrt{−\mathrm{1}}\Rightarrow\:{i}^{\mathrm{2}} =−\mathrm{1} \\ $$$${z}^{\frac{\mathrm{1}}{\mathrm{4}}} −{i}=\mathrm{0}\:\Rightarrow\:{z}^{\frac{\mathrm{1}}{\mathrm{4}}} ={i} \\ $$$$\Rightarrow\:{z}={i}^{\mathrm{4}} \:\Rightarrow\:{z}\:=\:\left({i}^{\mathrm{2}} \right)^{\mathrm{2}} \:\Rightarrow\:{z}\:=\:\left(−\mathrm{1}\right)^{\mathrm{2}} \:=\:\mathrm{1} \\ $$
Commented by Mastermind last updated on 10/Apr/22
$$ \\ $$$${So},\:{what}'{s}\:{the}\:{correct}\:{ans}.\:? \\ $$
Commented by MJS_new last updated on 10/Apr/22
$$\mathrm{the}\:\mathrm{correct}\:\mathrm{answer}\:\mathrm{is}:\:\mathrm{it}\:\mathrm{has}\:\mathrm{no}\:\mathrm{solution} \\ $$
Commented by MJS_new last updated on 10/Apr/22
$$\mathrm{but}\:\mathrm{1}^{\mathrm{1}/\mathrm{4}} =\mathrm{1}\neq\mathrm{i} \\ $$$$\mathrm{exponentiating}\:\mathrm{an}\:\mathrm{equation}\:\mathrm{might}\:\mathrm{introduce} \\ $$$$\mathrm{false}\:\mathrm{solutions} \\ $$$${z}^{\mathrm{1}/\mathrm{4}} =\mathrm{i}\:\mathrm{has}\:\mathrm{no}\:\mathrm{solution}. \\ $$$$\mathrm{similar}\:{z}^{\mathrm{1}/\mathrm{2}} =−\mathrm{1}\:\mathrm{has}\:\mathrm{no}\:\mathrm{solution}. \\ $$