%5E2.jpeg "(1-3i)^2")
Squaring Complex Numbers: A Look at (1 - 3i)^2
In the realm of complex numbers, squaring a complex number can seem daunting, but it's a straightforward process using the distributive property and the knowledge that i² = -1. Let's explore how to square (1 - 3i).
The Calculation
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Expansion: We start by expanding the expression: (1 - 3i)² = (1 - 3i) * (1 - 3i)
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Distribution: Applying the distributive property (FOIL method): (1 - 3i) * (1 - 3i) = 1 * (1 - 3i) - 3i * (1 - 3i)
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Simplifying: Expanding further: = 1 - 3i - 3i + 9i²
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Substituting i²: Replacing i² with -1: = 1 - 3i - 3i + 9(-1)
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Combining Like Terms: Simplifying the expression: = 1 - 6i - 9
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Final Result: = -8 - 6i
Conclusion
Therefore, the square of (1 - 3i) is -8 - 6i.
This process demonstrates that squaring a complex number involves both real and imaginary components. By understanding the fundamental properties of complex numbers, we can efficiently perform such operations.