(1+i)^5/(1-i)^3

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: (1+i)^5 / (1-i)^3

This article explores the process of simplifying the complex expression (1+i)^5 / (1-i)^3. We will leverage the properties of complex numbers and De Moivre's Theorem to arrive at a simplified solution.

Understanding Complex Numbers

A complex number is a number of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as √-1.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool for working with powers of complex numbers. It states that for any complex number in polar form z = r(cosθ + isinθ) and any integer 'n', the following holds:

z^n = r^n(cos(nθ) + isin(nθ))

Simplifying the Expression

1. Polar Form: We begin by expressing both (1+i) and (1-i) in polar form:

   • (1+i): Magnitude: |1+i| = √(1^2 + 1^2) = √2 Angle: θ = tan⁻¹(1/1) = π/4 Therefore, (1+i) = √2(cos(π/4) + isin(π/4))

   • (1-i): Magnitude: |1-i| = √(1^2 + (-1)^2) = √2 Angle: θ = tan⁻¹(-1/1) = -π/4 Therefore, (1-i) = √2(cos(-π/4) + isin(-π/4))

2. Applying De Moivre's Theorem:

   • (1+i)^5 = (√2)^5 (cos(5π/4) + isin(5π/4)) = 4√2 (-√2/2 - i√2/2) = -4 - 4i
   • (1-i)^3 = (√2)^3 (cos(-3π/4) + isin(-3π/4)) = 2√2 (-√2/2 - i√2/2) = -2 - 2i

3. Division: Now we can divide the results:

   (1+i)^5 / (1-i)^3 = (-4 - 4i) / (-2 - 2i)

4. Rationalizing the Denominator: To simplify further, we multiply both numerator and denominator by the complex conjugate of the denominator:

   (-4 - 4i) / (-2 - 2i) * (-2 + 2i) / (-2 + 2i) = (8 - 8i + 8i - 8i²) / (4 - 4i² )

5. Simplifying: Remembering that i² = -1, we get:

   (8 + 8) / (4 + 4) = 16 / 8 = 2

Final Result

Therefore, the simplified form of (1+i)^5 / (1-i)^3 is 2.