To compute \((x + iy)^3\), where \(x\) and \(iy\) are the real and imaginary parts respectively, we can expand it using the binomial theorem or by using the identity for complex powers.

### Step-by-Step Solution:

Let’s expand \((x + iy)^3\) directly:
\[
(x + iy)^3 = (x + iy)(x + iy)(x + iy)
\]
1. First, expand \((x + iy)^2\):
\[
(x + iy)^2 = x^2 + 2x(iy) + (iy)^2 = x^2 + 2ixy + i^2y^2
\]
Since \(i^2 = -1\), this simplifies to:
\[
x^2 + 2ixy - y^2
\]

2. Now multiply this result by \((x + iy)\):
\[
(x^2 + 2ixy - y^2)(x + iy)
\]

Expand this expression by distributing:
\[
= x(x^2 + 2ixy - y^2) + iy(x^2 + 2ixy - y^2)
\]

This gives:
\[
= x(x^2) + x(2ixy) + x(-y^2) + iy(x^2) + iy(2ixy) + iy(-y^2)
\]

Simplifying each term:
\[
= x^3 + 2ix^2y - xy^2 + ix^2y + 2i^2xy^2 - iy^3
\]
Since \(i^2 = -1\), the terms become:
\[
= x^3 + 2ix^2y - xy^2 + ix^2y - 2xy^2 - iy^3
\]

Now combine like terms:
\[
= (x^3 - 3xy^2) + i(3x^2y - y^3)
\]

### Final Answer:
\[
(x + iy)^3 = (x^3 - 3xy^2) + i(3x^2y - y^3)
\]

This is the expanded form of \((x + iy)^3\).

Would you like more details or have any questions?

### Related Questions:
1. How would you expand \((x + iy)^4\)?
2. What is the geometric interpretation of \((x + iy)^3\) in the complex plane?
3. How can De Moivre's Theorem help in expanding \((x + iy)^n\)?
4. What happens when you substitute \(x = 1\) and \(y = 1\) into the equation?
5. How can this be generalized for \((x + iy)^n\)?

### Tip:
Using De Moivre's Theorem can simplify powers of complex numbers in polar form!

Discover how to expand and simplify the complex expression (x + iy)³ using binomial expansion and key algebraic techniques. This solution explains the steps involved in obtaining the final form (x^3 - 3xy^2) + i(3x^2y - y^3), relevant for high school students studying complex numbers and algebra.

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