Find the value of (1 + i)^4.

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Question: Find the value of (1 + i)^4.

Options:

Correct Answer: 8

Solution:

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

Find the value of (1 + i)^4.

Find the value of (1 + i)^4.

Correct Answer: -4

Step 1: Recognize that (1 + i) can be expressed in polar form. The modulus (length) of (1 + i) is √2, and the argument (angle) is π/4.
Step 2: Write (1 + i) in polar form: (1 + i) = √2 e^(iπ/4).
Step 3: Raise (1 + i) to the power of 4: (1 + i)^4 = (√2 e^(iπ/4))^4.
Step 4: Use the property of exponents: (a^m)^n = a^(m*n). So, (√2)^4 = (√2)^4 = 2^2 = 4 and e^(iπ/4 * 4) = e^(iπ).
Step 5: Combine the results: (1 + i)^4 = 4 e^(iπ).
Step 6: Evaluate e^(iπ): e^(iπ) = -1 (from Euler's formula).
Step 7: Multiply the results: 4 * (-1) = -4.

Complex Numbers – Understanding the properties and operations involving complex numbers, particularly in polar form.
De Moivre's Theorem – Applying De Moivre's Theorem to raise complex numbers in polar form to a power.
Exponential Form of Complex Numbers – Converting complex numbers from rectangular to exponential form and vice versa.