Guys I think today we should discuss a really important problem \(i^{i}\). So please tell me where I am wrong. Note \(i\) denotes iota.

Let \[x=i^{i}\] \[x^ {4} = \sqrt{-1}^{4i}\] \[x^ {4}=1^{i}\] \[x^ {4}=1\] \[x=1\]

So comment and tell me where I am wrong

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## Comments

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TopNewest\(\displaystyle (i^i)^4 \neq (i^4)^i\)....playing with complex numbers is not so easy, instead you should start with taking log on both sides then evaluating it.

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Help me with this

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Let

\(\displaystyle x = i^i\)

Take log(natural logarithm) on both the sides and take the power down

\(\displaystyle ln(x) = i*ln(i)\)

Now recall that \(\displaystyle i = e^{i\frac{\pi}{2}}\) (\(\displaystyle e^{i\theta} = cos(\theta) + i*sin(\theta)\))

Substituting value of 'i' which is inside ln

\(\displaystyle ln(x) = i*ln(e^{i\frac{\pi}{2}})\)

Now take the power down to get (\(ln(e) = 1\))

\(\displaystyle ln(x) = i*(i\frac{\pi}{2})\)

Since \(i^2 = -1\) we get

\(\displaystyle ln(x) = -\frac{\pi}{2}\)

Now taking anti log to get

\[\displaystyle \boxed{x = e^{-\frac{\pi}{2}}}\]

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Apart from issues of complex exponentiation, bear in mind that whenever you manipulate an equation, you potentially introduce other solutions, especially if the statements are not "if and only if". See this for more examples.

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