Question
If $x = \sqrt{-1}$, then the value of $x^x$ is
The correct answer is
$e^{-\pi/2}$
To find the value of \(x^x\) where \(x = \sqrt{-1}\), we start by recognizing that \(x\) is the imaginary unit, \(i\) (i.e., \(i = \sqrt{-1}\)).
Express \(i\) in exponential form. We know from Euler's formula that:
- \(i = e^{i\pi/2}\)
Now, we need to find \(i^i\). Substitute \(i = e^{i\pi/2}\) into this expression:
\(i^i = (e^{i\pi/2})^i\)
By the power of a power property in exponents, we have:
- \((e^{i\pi/2})^i = e^{i \cdot (i\pi/2)} = e^{-\pi/2}\)
This is because multiplying \(i\pi/2\) by \(i\) gives \(-\pi/2\) (since \(i^2 = -1\)).
Thus, the value of \(x^x\) is \(e^{-\pi/2}\). This corresponds to option:
- \(e^{-\pi/2}\)
Therefore, the correct answer is \(e^{-\pi/2}\).
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Important Questions from Complex Numbers
- Consider the following complex numbers
$$ z_1 = r_1(\cos \theta_1 + i \, \sin \theta_1) $$
$$ z_2 = r_2(\cos \theta_2 + i \, \sin \theta_2) $$
where $r_1, r_2$ are real numbers, $0 \leq \theta_1 \leq \frac{\pi}{2}$, $0 \leq \theta_2 \leq \frac{\pi}{2}$, and $i = \sqrt{-1}$
If $|z_1 + z_2| = |z_1| + |z_2|$, which one of the following conditions is necessarily CORRECT? - What is the value of \((2 - 3i) + (3 + 5i)\)
- What is the real part of the number \(15 + 2i\)
- What is the real part of the number $5i$?
- What is the value of $(6 + 3i) + (2 + 5i)$?
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