To solve the expression \( \frac{(1 + i)^8}{(1 - i)^6} \) and determine its correct representation, we will utilize Euler's formula and properties of complex numbers.

1. Express \( 1 + i \) and \( 1 - i \) in polar form:
   • \(1 + i = \sqrt{2} \, e^{i \frac{\pi}{4}}\)
   • \(1 - i = \sqrt{2} \, e^{-i \frac{\pi}{4}}\)
2. Substitute these into the expression:

\(\frac{(1 + i)^8}{(1 - i)^6} = \frac{(\sqrt{2} \, e^{i \frac{\pi}{4}})^8}{(\sqrt{2} \, e^{-i \frac{\pi}{4}})^6}\)

1. Simplify the numerator and the denominator:
   • Numerator: \( (\sqrt{2})^8 \, (e^{i \frac{\pi}{4}})^8 = 16 \, e^{i 2\pi} = 16 \)
   • Denominator: \( (\sqrt{2})^6 \, (e^{-i \frac{\pi}{4}})^6 = 8 \, e^{-i \frac{3\pi}{2}} = 8 \, e^{i \frac{\pi}{2}} \)
2. Compute the final expression:

\(\frac{16}{8} \cdot \left( \frac{1}{e^{i \frac{\pi}{2}}} \right) = 2 \, e^{-i \frac{\pi}{2}}\)

1. Thus, the expression simplifies to \(2 \, e^{-i \frac{\pi}{2}}\).

This matches option:

$2e^{-i\frac{\pi}{2}}$

1. .

Therefore, the correct representation of the expression is $2e^{-i\frac{\pi}{2}}$, making it the correct answer.