If z = re^(iθ), what is the value of z^2?
Correct Answer: r^2e^(i2θ)
- Step 1: Start with the given expression for z, which is z = re^(iθ).
- Step 2: To find z^2, we need to square the expression for z.
- Step 3: Write z^2 as (re^(iθ))^2.
- Step 4: Use the property of exponents that states (a*b)^n = a^n * b^n.
- Step 5: Apply this property: (re^(iθ))^2 = r^2 * (e^(iθ))^2.
- Step 6: Now, square e^(iθ) using the property (e^(iθ))^2 = e^(i*2θ).
- Step 7: Combine the results: z^2 = r^2 * e^(i*2θ).
- Step 8: Write the final result as z^2 = r^2e^(i2θ).
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