Question

Imaginary number to the power of another imaginary number is real .
What is the value of i^i?
Explain how u get the value
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Answer 1

We know that i=−1i=\sqrt{-1}i=−1​ and that i2=−1i^2=-1i2=−1.

We know that i3=i2⋅ii^3=i^2\cdot ii3=i2⋅i. But since i2=−1{i^2=-1}i2=−1, we see that:

   i3=i2⋅i=(−1)⋅i=−i\begin{aligned} i^3 &= {{i^2}}\cdot i\\ \\ &={ (-1)}\cdot i\\ \\ &= \purpleD{-i} \end{aligned}i3​=i2⋅i=(−1)⋅i=−i​

Similarly i4=i2⋅i2i^4=i^2\cdot i^2i4=i2⋅i2. Again, using the fact that i2=−1{i^2=-1}i2=−1, we have the following:

   i4=i2⋅i2=(−1)⋅(−1)=1\begin{aligned} i^4 &= {{i^2\cdot i^2}}\\ \\ &=({ -1})\cdot ({-1})\\ \\ &= \goldD{1} \end{aligned}i4​=i2⋅i2=(−1)⋅(−1)=1​

Answer 2

I have proved myself ....I don't know ...is it correct check it ok