Start with z = x + yi.
Compute z³ = (x + yi)³.
Expand using the binomial theorem:
(x + yi)³ = x³ + 3x²(yi) + 3x(yi)² + (yi)³
Expand and simplify each term:
= x³ + 3x²yi + 3xy²(i²) + y³(i³)
Knowing i² = -1 and i³ = -i, substitute to simplify:
= x³ + 3x²yi – 3xy² – y³i
Now, factor out the i in the imaginary terms:
= x³ – 3xy² + i(3x²y – y³)
Group the real and imaginary parts:
Real part: x³ – 3xy²
Imaginary part: i(3x²y – y³)
So, z³ = (x³ – 3xy²) + i(3x²y – y³).
Find the complex conjugate of each number below quiz.
Compute z³ = (x + yi)³.
Expand using the binomial theorem:
(x + yi)³ = x³ + 3x²(yi) + 3x(yi)² + (yi)³
Expand and simplify each term:
= x³ + 3x²yi + 3xy²(i²) + y³(i³)
Knowing i² = -1 and i³ = -i, substitute to simplify:
= x³ + 3x²yi – 3xy² – y³i
Now, factor out the i in the imaginary terms:
= x³ – 3xy² + i(3x²y – y³)
Group the real and imaginary parts:
Real part: x³ – 3xy²
Imaginary part: i(3x²y – y³)
So, z³ = (x³ – 3xy²) + i(3x²y – y³).