Complex Number Exponentiation: Finding the Value of x^x

This problem involves understanding the imaginary unit and applying Euler's formula to evaluate a complex number raised to a complex power. We are given the value of x and asked to find x^x.

Understanding the Imaginary Unit (x)

The problem states that \(x = \sqrt{-1}\). In mathematics, \(\sqrt{-1}\) is defined as the imaginary unit, commonly denoted by the symbol i. Therefore, we have \(x = i\).

Now, the task is to calculate the value of \(x^x\), which translates to calculating \(i^i\).

Euler's Formula and Polar Form of Complex Numbers

To evaluate \(i^i\), we need to express the imaginary unit i in its exponential form. This can be done using Euler's formula, which states:

\[e^{i\theta} = \cos(\theta) + i\sin(\theta)\]

For the imaginary unit \(i\), we know that \(i = 0 + 1i\). We need to find an angle \(\theta\) such that \(\cos(\theta) = 0\) and \(\sin(\theta) = 1\).

This condition is satisfied when \(\theta = \frac{\pi}{2}\) (or \(\frac{\pi}{2} + 2n\pi\), where \(n\) is an integer. For the principal value, we use \(n=0\)).

Substituting \(\theta = \frac{\pi}{2}\) into Euler's formula:

\[e^{i\frac{\pi}{2}} = \cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right)\]

\[e^{i\frac{\pi}{2}} = 0 + i(1)\]

\[e^{i\frac{\pi}{2}} = i\]

So, the imaginary unit \(i\) can be expressed in exponential form as \(e^{i\frac{\pi}{2}}\).

Calculating \(x^x\) (iⁱ) Step-by-Step

Now that we have the exponential form of \(i\), we can substitute it into the expression \(i^i\):

We want to find \(i^i\).

Substitute \(i = e^{i\frac{\pi}{2}}\):

\[i^i = \left(e^{i\frac{\pi}{2}}\right)^i\]

Using the exponent rule \((a^b)^c = a^{bc}\), we multiply the exponents:

\[i^i = e^{i \cdot \left(i\frac{\pi}{2}\right)}\]

Simplify the product of \(i\) terms in the exponent:

\[i \cdot i = i^2\]

We know that \(i^2 = -1\).

Substitute \(i^2 = -1\) back into the expression:

\[i^i = e^{-1 \cdot \frac{\pi}{2}}\]

\[i^i = e^{-\frac{\pi}{2}}\]

Final Value of the Expression

Based on our step-by-step calculation, the value of \(x^x\) when \(x = \sqrt{-1}\) is \(e^{-\frac{\pi}{2}}\).

Therefore, the value of \(x^x\) is \(e^{-\pi/2}\).