5Question: An AI-powered medical imaging system processes a signal modeled by the complex number $ z = \cos\left(\frac{\pi}{6}\right) + i \sin\left(\frac{\pi}{6}\right) $. Compute $ z^6 $ and express it in the form $ a + bi $.
["# Using Euler’s Formula to Compute $ z^6 $ in Medical Imaging Signal Analysis", "In modern medical imaging systems powered by artificial intelligence, signal processing plays a critical role in enhancing image clarity and enabling accurate diagnostics. One key mathematical operation involves analyzing complex-valued signals, often modeled as complex numbers. This article explores how an AI-powered medical imaging system processes a signal represented by the complex number\n$$\nz = \cos\left(\frac{\pi}{6}\right) + i \sin\left(\frac{\pi}{6}\right),\n$$\nand computes $ z^6 $ using powerful tools from complex analysis.", "## Understanding the Input Signal", "The given complex number is in trigonometric form:\n$$\nz = \cos\left(\frac{\pi}{6}\right) + i \sin\left(\frac{\pi}{6}\right).\n$$\nThis expression aligns with Euler’s formula:\n$$\ne^{i\ heta} = \cos\ heta + i\sin\ heta.\n$$\nThus, we can rewrite $ z $ as:\n$$\nz = e^{i\pi/6}.\n$$", "## Computing $ z^6 $", "Using the exponential form:\n$$\nz^6 = \left(e^{i\pi/6}\right)^6 = e^{i\cdot 6 \cdot \frac{\pi}{6}} = e^{i\pi}.\n$$\nNow, apply Euler’s formula again:\n$$\ne^{i\pi} = \cos\pi + i\sin\pi = -1 + i\cdot 0 = -1.\n$$", "## Final Result", "Therefore, the computed value of $ z^6 $ is:\n$$\n\boxed{-1 + 0i} \quad \ ext{or simply} \quad \boxed{-1}.\n$$", "This result demonstrates how AI-driven systems leverage complex number arithmetic to transform raw biomedical signals into meaningful coefficients used in pattern recognition and diagnostic modeling. Understanding such mathematical foundations enables improved algorithm design and deeper insight into signal behavior in medical applications.", "This concise yet powerful computation illustrates why AI systems in medical imaging rely on robust numerical foundations—turning trigonometric signals into interpretable real-world outputs."]
