If z = x + yi, find the real part of z^3.

If z = x + yi, find the real part of z^3.

Step 1: Start with the expression for z, which is z = x + yi, where x is the real part and yi is the imaginary part.
Step 2: We need to find z^3, which means we will calculate (x + yi)^3.
Step 3: Use the binomial expansion formula, which states (a + b)^n = sum of (n choose k) * a^(n-k) * b^k for k from 0 to n.
Step 4: In our case, a = x, b = yi, and n = 3. So we will expand (x + yi)^3.
Step 5: Calculate each term in the expansion: (x + yi)^3 = x^3 + 3(x^2)(yi) + 3(x)(yi)^2 + (yi)^3.
Step 6: Simplify each term: (yi)^2 = -y^2 and (yi)^3 = -y^3i. So we have x^3 + 3x^2(yi) - 3xy^2 - y^3i.
Step 7: Combine the real parts and the imaginary parts: Real part = x^3 - 3xy^2 and Imaginary part = (3x^2y - y^3)i.
Step 8: The real part of z^3 is x^3 - 3xy^2.

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