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

solve z^(1/4) −i=0 Mastermind

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

Answered by nurtani last updated on 10/Apr/22

i=(√(−1))⇒ i^2 =−1 z^(1/4) −i=0 ⇒ z^(1/4) =i ⇒ z=i^4 ⇒ z = (i^2 )^2 ⇒ z = (−1)^2 = 1

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

Commented by Mastermind last updated on 10/Apr/22

So, what′s the correct ans. ?

$$ \\ $$$${So},\:{what}'{s}\:{the}\:{correct}\:{ans}.\:? \\ $$

Commented by MJS_new last updated on 10/Apr/22

the correct answer is: it has no solution

$$\mathrm{the}\:\mathrm{correct}\:\mathrm{answer}\:\mathrm{is}:\:\mathrm{it}\:\mathrm{has}\:\mathrm{no}\:\mathrm{solution} \\ $$

Commented by MJS_new last updated on 10/Apr/22

but 1^(1/4) =1≠i exponentiating an equation might introduce false solutions z^(1/4) =i has no solution. similar z^(1/2) =−1 has no solution.

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

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