Step 1: Start with the given value of z, which is z = e^(iπ/4).
Step 2: To find z^8, we need to raise z to the power of 8. This means we calculate (e^(iπ/4))^8.
Step 3: When raising a power to another power, we multiply the exponents. So, (e^(iπ/4))^8 = e^(i(π/4) * 8).
Step 4: Calculate the exponent: (π/4) * 8 = 2π. So now we have e^(i2π).
Step 5: Recall that e^(i2π) is a well-known result in complex numbers, which equals 1.
Step 6: Therefore, z^8 = 1.
Complex Exponentiation – Understanding how to manipulate complex numbers in exponential form, particularly using Euler's formula.
Properties of Exponents – Applying the rules of exponents to simplify expressions involving powers.
Unit Circle and Trigonometry – Recognizing that e^(iθ) corresponds to points on the unit circle, and understanding periodicity in complex exponentials.
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