The discussion focuses on calculating \( z^5 \) where \( z = e^{(2-\frac{i \pi}{4})} \). The correct approach involves applying the power rule for exponents, resulting in \( z^5 = e^{5(2-\frac{i \pi}{4})} \). This method simplifies the calculation without the need for expansion, demonstrating the efficiency of using exponential properties in complex number simplification.
PREREQUISITESMathematicians, physics students, and anyone interested in complex analysis or exponential functions will benefit from this discussion.
Guest said:If $z = e^{(2-\frac{i \pi}{4})}$ what's $z^5$?
The only way I can think of doing this is expanding $(2-\frac{i \pi}{4})^5$, but I think I'm supposed to use a simpler method (not sure what it's).
Thank you, I like Serena. I get it now. (Smile)I like Serena said:Hi Guest, (Smile)
Let's substitute and apply a power rule:
$$z^5=\left(e^{(2-\frac{i \pi}{4})}\right)^5
=e^{5(2-\frac{i \pi}{4})}
$$