If z = 1 + i, find z^3.

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Question: If z = 1 + i, find z^3.

Options:

Correct Answer: -2 + 2i

Solution:

z^3 = (1 + i)^3 = 1 + 3i - 3 - i = -2 + 2i.

If z = 1 + i, find z^3.

If z = 1 + i, find z^3.

Correct Answer: -2 + 2i

Step 1: Start with the expression z = 1 + i.
Step 2: We need to find z^3, which means we will calculate (1 + i)^3.
Step 3: Use the binomial expansion formula: (a + b)^n = a^n + n*a^(n-1)*b + ... + b^n.
Step 4: Here, a = 1, b = i, and n = 3. So we calculate (1 + i)^3.
Step 5: Calculate each term: 1^3 = 1, 3 * 1^2 * i = 3i, 3 * 1 * i^2 = 3 * i^2, and i^3 = i * i^2.
Step 6: Remember that i^2 = -1. So, 3 * i^2 = 3 * (-1) = -3.
Step 7: Now, combine all the terms: 1 + 3i - 3 + 0i.
Step 8: Combine like terms: (1 - 3) + (3i + 0i) = -2 + 3i.
Step 9: Therefore, z^3 = -2 + 3i.

Complex Numbers – Understanding the properties and operations involving complex numbers, including addition, multiplication, and exponentiation.
Binomial Expansion – Applying the binomial theorem to expand expressions raised to a power, particularly for complex numbers.
Algebraic Manipulation – Performing algebraic operations accurately, including combining like terms and simplifying expressions.