To determine which expression is equivalent to \(\left(\frac{1}{2}\left[\cos \left(\frac{\pi}{5}\right) + i \sin \left(\frac{\pi}{5}\right)\right]\right)^5\), we need to follow the steps of calculating the modulus and argument of the initial expression and then raising these to the power of 5.
1. Initial Expression:
[tex]\[
z = \frac{1}{2}\left(\cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right)\right)
\][/tex]
Here, \(z\) is a complex number in polar form, where the modulus \(r\) is \(\frac{1}{2}\) and the argument \(\theta\) is \(\frac{\pi}{5}\).
2. Modulus and Argument:
The modulus of \(z\) is \(\frac{1}{2}\), and the argument of \(z\) is \(\frac{\pi}{5}\).
3. Raising to the 5th Power:
We need to raise both the modulus and argument to the 5th power for the given operation \(\left(z\right)^5\):
- The new modulus \(r^{\prime} = \left(\frac{1}{2}\right)^5 = \frac{1}{32}\).
- The new argument \(\theta^{\prime} = 5 \times \frac{\pi}{5} = \pi\).
4. Equivalent Expression:
Rewriting the complex number in polar form with the new modulus and argument:
[tex]\[
\left(z\right)^5 = \frac{1}{32} \left[\cos(\pi) + i \sin(\pi)\right]
\][/tex]
Therefore, the equivalent expression is:
[tex]\[
\boxed{\frac{1}{32}[\cos (\pi) + i \sin (\pi)]}
\}
Thus, the correct answer is:
\[
\frac{1}{32} \left[\cos (\pi) + i \sin (\pi)\right]
\][/tex]
