If z = 1 + i, what is the value of z^3? (2023)

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Question: If z = 1 + i, what is the value of z^3? (2023)

Options:

Correct Answer: -2 + 2i

Exam Year: 2023

Solution:

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

If z = 1 + i, what is the value of z^3? (2023)

If z = 1 + i, what is the value of z^3? (2023)

Step 1: Start with the given value of z, which is z = 1 + i.
Step 2: We need to calculate z^3, which means we need to find (1 + i)^3.
Step 3: Use the binomial expansion formula: (a + b)^n = a^n + n*a^(n-1)*b + (n*(n-1)/2)*a^(n-2)*b^2 + ... + b^n.
Step 4: In our case, a = 1, b = i, and n = 3. So we expand (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^3.
Step 6: Recall that i^2 = -1 and i^3 = -i. So, replace i^2 and i^3 in the expression.
Step 7: Now we have: 1 + 3i + 3(-1) + (-i).
Step 8: Simplify the expression: 1 + 3i - 3 - i.
Step 9: Combine like terms: (1 - 3) + (3i - i) = -2 + 2i.
Step 10: Therefore, the value of z^3 is -2 + 2i.

Complex Numbers – Understanding the properties and operations involving complex numbers, including addition, multiplication, and exponentiation.
Binomial Expansion – Applying the binomial theorem to expand expressions of the form (a + b)^n.
Powers of i – Knowing the cyclical nature of the powers of the imaginary unit i, where i^2 = -1, i^3 = -i, and i^4 = 1.