To compute $(x + iy)^2(x - iy)^2$, we will proceed step by step.

Step 1: Expand $(x + iy)^2$

Using the formula $(a + b)^2 = a^2 + 2ab + b^2$: $(x + iy)^2 = x^2 + 2x(iy) + (iy)^2$ $= x^2 + 2ixy - y^2$

Thus, $(x + iy)^2 = x^2 - y^2 + 2ixy$

Step 2: Expand $(x - iy)^2$

Similarly, expanding $(x - iy)^2$: $(x - iy)^2 = x^2 - 2x(iy) + (iy)^2$ $= x^2 - 2ixy - y^2$

Thus, $(x - iy)^2 = x^2 - y^2 - 2ixy$

Step 3: Multiply $(x + iy)^2$ and $(x - iy)^2$

Now, multiply the two results: $(x + iy)^2(x - iy)^2 = (x^2 - y^2 + 2ixy)(x^2 - y^2 - 2ixy)$

Step 4: Use the difference of squares formula

This is a difference of squares: $(a + b)(a - b) = a^2 - b^2$ Here, $a = x^2 - y^2$ and $b = 2ixy$. Substituting these: $(x + iy)^2(x - iy)^2 = (x^2 - y^2)^2 - (2ixy)^2$

Step 5: Simplify the terms

1. Compute $(x^2 - y^2)^2$: $(x^2 - y^2)^2 = (x^2 - y^2)(x^2 - y^2)$
2. Compute $(2ixy)^2$: $(2ixy)^2 = (2xy)^2 \cdot i^2 = 4x^2y^2 \cdot (-1) = -4x^2y^2$

Substitute these back into the equation: $(x + iy)^2(x - iy)^2 = (x^2 - y^2)^2 - (-4x^2y^2)$ $(x + iy)^2(x - iy)^2 = (x^2 - y^2)^2 + 4x^2y^2$

Step 6: Final expression

This is the final simplified result: $(x + iy)^2(x - iy)^2 = (x^2 + y^2)^2$

Conclusion

The final result is: $(x + iy)^2(x - iy)^2 = (x^2 + y^2)^2$

Would you like more details or have any questions?

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Related Questions:

1. What is the geometric interpretation of $(x^2 + y^2)$ in the complex plane?
2. How would the result change if $i$ was replaced with a general complex constant $z$?
3. What happens if we expand $(x + iy)^3(x - iy)^3$?
4. Can we factorize $(x^2 + y^2)^2$ further?
5. How is this computation related to modulus of a complex number?

Tip:

When working with complex numbers, always check for symmetry properties (like conjugates) to simplify calculations.