If z = 1 + i, find the value of z^4.

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Question: If z = 1 + i, find the value of z^4.

Options:

Correct Answer: -4

Solution:

z^4 = (1 + i)^4 = (2e^(iπ/4))^4 = 16e^(iπ) = -16.

If z = 1 + i, find the value of z^4.

If z = 1 + i, find the value of z^4.

Correct Answer: -16

Step 1: Identify the value of z. Here, z = 1 + i.
Step 2: To find z^4, we need to calculate (1 + i)^4.
Step 3: Convert 1 + i into polar form. The magnitude (r) is √(1^2 + 1^2) = √2.
Step 4: Find the angle (θ). The angle is arctan(1/1) = π/4.
Step 5: Write the polar form of z. It is 2e^(iπ/4) because r = √2 and we multiply by √2 to get 2.
Step 6: Now raise the polar form to the power of 4: (2e^(iπ/4))^4.
Step 7: Apply the power to both the magnitude and the angle: 2^4 * e^(i(4 * π/4)).
Step 8: Calculate 2^4 = 16 and 4 * π/4 = π, so we have 16e^(iπ).
Step 9: Evaluate e^(iπ). According to Euler's formula, e^(iπ) = -1.
Step 10: Multiply 16 by -1 to get the final result: 16 * -1 = -16.

Complex Numbers – Understanding the properties and operations of complex numbers, including exponentiation and polar form.
De Moivre's Theorem – Applying De Moivre's Theorem to compute powers of complex numbers in polar form.