If z = re^(iθ), what is the value of z^3?

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Q: If z = re^(iθ), what is the value of z^3?

Solution: Using the properties of exponents, z^3 = (re^(iθ))^3 = r^3 e^(i3θ).

Steps: 7

Step 1: Start with the given expression for z, which is z = re^(iθ).
Step 2: To find z^3, we need to raise z to the power of 3, so we write z^3 = (re^(iθ))^3.
Step 3: Apply the property of exponents that states (a*b)^n = a^n * b^n. Here, a = r and b = e^(iθ), and n = 3.
Step 4: This means we can separate the terms: (re^(iθ))^3 = r^3 * (e^(iθ))^3.
Step 5: Now, calculate (e^(iθ))^3 using the property of exponents: (e^(iθ))^3 = e^(i3θ).
Step 6: Combine the results from Step 4 and Step 5: z^3 = r^3 * e^(i3θ).
Step 7: Therefore, the final result is z^3 = r^3 e^(i3θ).