If z = 1 + i, find the value of z^3.

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

Options:

Correct Answer: -2 + 2i

Solution:

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

If z = 1 + i, find the value of z^3.

If z = 1 + i, find the value of z^3.

Correct Answer: -2 + 2i

Step 1: Start with the value of z, which is 1 + i.
Step 2: To find z^3, we need to calculate (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 + ...
Step 4: Here, a = 1, b = i, and n = 3. So we calculate: (1 + i)^3 = 1^3 + 3*1^2*i + 3*1*i^2 + i^3.
Step 5: Calculate each term: 1^3 = 1, 3*1^2*i = 3i, 3*1*i^2 = 3*(-1) = -3, and i^3 = -i.
Step 6: Now combine all the terms: 1 + 3i - 3 - i.
Step 7: Combine like terms: (1 - 3) + (3i - i) = -2 + 2i.
Step 8: Therefore, z^3 = -2 + 2i.

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