Understanding the Problem

The question asks for the value of the complex number expression $i^{-1/2}$, where $i$ is the imaginary unit ($i = \sqrt{-1}$). We need to find which of the given options matches the calculated value.

Mathematical Approach

To evaluate expressions like $i^x$, it's often easiest to use the polar form of the complex number $i$. The polar form of $i$ is $e^{i\pi/2}$.

Calculation Steps

1. Express $i$ in polar form: $i = \cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2}) = e^{i\pi/2}$.
2. Apply the exponent: We need to calculate $i^{-1/2}$. Using the polar form: $i^{-1/2} = (e^{i\pi/2})^{-1/2}$ Using the rule $(a^m)^n = a^{m \times n}$: $i^{-1/2} = e^{i(\pi/2 \times -1/2)} = e^{i(-\pi/4)}$
3. Convert back to rectangular form (a + bi): Using Euler's formula ($e^{i\theta} = \cos(\theta) + i\sin(\theta)$): $e^{i(-\pi/4)} = \cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4})$
4. Evaluate trigonometric functions: Recall the values for the angle $-\frac{\pi}{4}$ (or $-45^\circ$): $\cos(-\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$ $\sin(-\frac{\pi}{4}) = -\sin(\frac{\pi}{4}) = -\frac{1}{\sqrt{2}}$
5. Combine the results: Substitute the evaluated trigonometric values back into the expression: $i^{-1/2} = \frac{1}{\sqrt{2}} + i(-\frac{1}{\sqrt{2}})$ $i^{-1/2} = \frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}}$ Factor out $\frac{1}{\sqrt{2}}$: $i^{-1/2} = \frac{1}{\sqrt{2}}(1-i)$

Final Answer Verification

The calculated value is $\frac{1}{\sqrt{2}}(1-i)$. This matches the first option provided.