If z = 2 + 2i, find the value of z^3.

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Question: If z = 2 + 2i, find the value of z^3.

Options:

Correct Answer: -8 + 8i

Solution:

z^3 = (2 + 2i)^3 = 8 + 12i - 12 - 8i = -8 + 4i.

If z = 2 + 2i, find the value of z^3.

If z = 2 + 2i, find the value of z^3.

Correct Answer: -8 + 4i

Step 1: Identify the value of z, which is 2 + 2i.
Step 2: To find z^3, we need to calculate (2 + 2i)^3.
Step 3: Use the binomial expansion formula: (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.
Step 4: Here, a = 2 and b = 2i. Calculate a^3: 2^3 = 8.
Step 5: Calculate 3a^2b: 3 * (2^2) * (2i) = 3 * 4 * 2i = 24i.
Step 6: Calculate 3ab^2: 3 * (2) * (2i)^2 = 3 * 2 * (-4) = -24.
Step 7: Calculate b^3: (2i)^3 = 8i^3 = 8 * (-1) = -8.
Step 8: Now combine all the parts: 8 + 24i - 24 - 8.
Step 9: Combine the real parts: 8 - 24 - 8 = -24.
Step 10: Combine the imaginary parts: 24i.
Step 11: The final result is -24 + 24i.

Complex Numbers – Understanding the properties and operations of complex numbers, including addition, multiplication, and exponentiation.
Binomial Expansion – Applying the binomial theorem or expansion techniques to calculate powers of binomials.
Imaginary Unit – Recognizing and correctly handling the imaginary unit 'i', where i^2 = -1.