# If \[i{z^4} + 1 = 0\], then $z$ can take the value of

A. $\frac{{1 + i}}{{\sqrt 2 }}$

B. $\cos \frac{\pi }{8} + i\sin \frac{\pi }{8}$

C. $\frac{1}{{4i}}$

D. $i$

Answer

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363.9k+ views

**Hint:**To find the value of $z$ we are going to simplify the given equation and bring it in terms of $z$.

Complete step by step answer:

Complete step by step answer:

The given equation can be written as,

$i{z^4} = - 1$

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

${z^4} = i$

If we simplify the above equation, we get,

$z = {\left( i \right)^{1/4}} = {\left( {0 + i} \right)^{^{1/4}}}$

Option B is the correct answer for this question.

**In the above equation we added zero in the front to get it into the general form of complex number.**

Note:

Note:

Therefore,

$z = {\left( {\cos \frac{\pi }{2} + i\sin \frac{\pi }{2}} \right)^{1/4}}$

$z = \cos \frac{\pi }{8} + i\sin \frac{\pi }{8}$

Last updated date: 23rd Sep 2023

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