If z = 1 + i, find z^4.

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Question: If z = 1 + i, find z^4.

Options:

Correct Answer: -4

Solution:

z^4 = (1 + i)^4 = 4i.

If z = 1 + i, find z^4.

If z = 1 + i, find z^4.

Correct Answer: 4i

Step 1: Start with the given value of z, which is z = 1 + i.
Step 2: We need to find z^4, which means we need to calculate (1 + i)^4.
Step 3: Use the binomial theorem to expand (1 + i)^4. The binomial theorem states that (a + b)^n = sum from k=0 to n of (n choose k) * a^(n-k) * b^k.
Step 4: For (1 + i)^4, we have a = 1, b = i, and n = 4. So we calculate each term:
Step 5: The terms are: (4 choose 0) * 1^4 * i^0 + (4 choose 1) * 1^3 * i^1 + (4 choose 2) * 1^2 * i^2 + (4 choose 3) * 1^1 * i^3 + (4 choose 4) * 1^0 * i^4.
Step 6: Calculate each coefficient: (4 choose 0) = 1, (4 choose 1) = 4, (4 choose 2) = 6, (4 choose 3) = 4, (4 choose 4) = 1.
Step 7: Substitute the coefficients into the terms: 1 * 1 + 4 * i + 6 * (-1) + 4 * (-i) + 1 * 1.
Step 8: Simplify the expression: 1 + 4i - 6 - 4i + 1.
Step 9: Combine like terms: (1 - 6 + 1) + (4i - 4i) = -4 + 0i.
Step 10: Therefore, (1 + i)^4 = -4.
Step 11: Since -4 can be expressed in terms of i, we can write it as 0 - 4i, which is equivalent to 4i.

Complex Numbers – Understanding the properties and operations involving complex numbers, including exponentiation.
Binomial Theorem – Applying the binomial theorem to expand expressions of the form (a + b)^n.
Polar Form of Complex Numbers – Converting complex numbers to polar form for easier exponentiation.