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If z = re^(iθ), what is the value of z^2?

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Question: If z = re^(iθ), what is the value of z^2?

Options:

  1. r^2e^(i2θ)
  2. re^(iθ)
  3. 2re^(iθ)
  4. r^2e^(iθ)

Correct Answer: r^2e^(i2θ)

Solution: 

Using the property of exponents, z^2 = (re^(iθ))^2 = r^2e^(i2θ).

If z = re^(iθ), what is the value of z^2?

Practice Questions

Q1
If z = re^(iθ), what is the value of z^2?
  1. r^2e^(i2θ)
  2. re^(iθ)
  3. 2re^(iθ)
  4. r^2e^(iθ)

Questions & Step-by-Step Solutions

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).
  • Complex Numbers – Understanding the representation of complex numbers in polar form and the properties of exponents.
  • Exponential Form – Using Euler's formula to manipulate complex numbers expressed in exponential form.
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