%5E3(1%2Bi).jpeg "(1-i)^3(1+i)")
Simplifying Complex Expressions: (1 - i)³ (1 + i)
This article will explore the simplification of the complex expression (1 - i)³ (1 + i). We will use the properties of complex numbers and algebraic manipulation to arrive at a simplified form.
Understanding Complex Numbers
Complex numbers are expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1 (i² = -1).
Simplifying the Expression
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Expanding the Cube: We begin by expanding the cube of (1 - i):
(1 - i)³ = (1 - i)(1 - i)(1 - i)
Expanding the first two factors:
(1 - i)(1 - i) = 1 - i - i + i² = 1 - 2i - 1 = -2i
Now, multiplying this result by (1 - i):
(-2i)(1 - i) = -2i + 2i² = -2i - 2 = -2 - 2i
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Multiplying by (1 + i):
We now have (-2 - 2i) multiplied by (1 + i):
(-2 - 2i)(1 + i) = -2 - 2i - 2i - 2i² = -2 - 4i + 2 = -4i
Conclusion
By using the properties of complex numbers and algebraic manipulation, we have simplified the expression (1 - i)³ (1 + i) to -4i. This demonstrates the power of complex number operations and their application in simplifying complex expressions.