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Question Number 168399 by Mastermind last updated on 09/Apr/22

$$solve$$$$z^{\frac{1}{4}}-i=0$$$$$$$$Mastermind$$

Answered by nurtani last updated on 10/Apr/22

$$i=\sqrt{-1}\Rightarrow i^2=-1$$$$z^{\frac{1}{4}}-i=0\Rightarrow z^{\frac{1}{4}}=i$$$$\Rightarrow z=i^4\Rightarrow z={\left(i^2\right)}^2\Rightarrow z={(-1)}^2=1$$

Commented by Mastermind last updated on 10/Apr/22

$$$$$$So,{what}^{\prime }sthecorrectans.?$$

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

$$\mathrm{but}{1}^{1/4}=1\ne \mathrm{i}$$$$\mathrm{exponentiating}\mathrm{an}\mathrm{equation}\mathrm{might}\mathrm{introduce}$$$$\mathrm{false}\mathrm{solutions}$$$$z^{1/4}=\mathrm{i}\mathrm{has}\mathrm{no}\mathrm{solution}.$$$$\mathrm{similar}{z}^{1/2}=-1\mathrm{has}\mathrm{no}\mathrm{solution}.$$

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