%5E3%2F1-i%5E3.jpeg "(1-i)^3/1-i^3")
Simplifying Complex Expressions: (1-i)^3 / (1-i^3)
This article explores the simplification of the complex expression (1-i)^3 / (1-i^3). We will utilize fundamental properties of complex numbers and algebraic manipulations 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 (i.e., i^2 = -1).
Simplifying the Expression
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Expanding the Numerator: We begin by expanding the numerator, (1-i)^3: (1-i)^3 = (1-i)(1-i)(1-i) = (1-2i+i^2)(1-i) = (-2i)(1-i) = -2i + 2i^2 = -2i - 2
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Simplifying the Denominator: Next, let's simplify the denominator, (1-i^3): Since i^2 = -1, we have: 1 - i^3 = 1 - (i^2 * i) = 1 - (-1 * i) = 1 + i
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Combining the Simplified Terms: Now, we can combine the simplified numerator and denominator: (1-i)^3 / (1-i^3) = (-2i - 2) / (1 + i)
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Rationalizing the Denominator: To eliminate the complex number in the denominator, we multiply both numerator and denominator by the conjugate of the denominator, which is (1 - i): (-2i - 2) / (1 + i) * (1 - i) / (1 - i) = (-2i + 2i^2 - 2 + 2i) / (1 - i^2) = (-2 - 2) / (1 + 1) = -4 / 2 = -2
Final Result
Therefore, the simplified form of the expression (1-i)^3 / (1-i^3) is -2.