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

Jasonbradley

3 min read

·

Jun 16, 2024

23

Share

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

This article delves into simplifying the complex expression (1+i/1-i)^3-(1-i/1+i)^3. We'll utilize fundamental concepts of complex numbers and algebraic manipulation to arrive at a neat and concise 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 the square root of -1.

Simplifying the Expression

Let's break down the expression step by step:

1. Rationalizing the denominators:

   • (1+i/1-i) can be simplified by multiplying both numerator and denominator by the conjugate of the denominator (1+i):

   (1+i/1-i) * (1+i/1+i) = (1 + 2i + i^2) / (1 - i^2) = (2i)/2 = i

   • Similarly, (1-i/1+i) can be simplified by multiplying by (1-i):

   (1-i/1+i) * (1-i/1-i) = (1 - 2i + i^2) / (1 - i^2) = (-2i)/2 = -i

2. Substituting the simplified values:

   Our expression now becomes: (i)^3 - (-i)^3

3. Evaluating the powers:

   • i^3 = i^2 * i = -1 * i = -i
   • (-i)^3 = (-1)^3 * i^3 = -1 * -i = i

4. Final Calculation:

   The expression now simplifies to: -i - i = -2i

Conclusion

By rationalizing the denominators and applying the properties of complex numbers, we have successfully simplified the expression (1+i/1-i)^3-(1-i/1+i)^3 to -2i. This demonstrates how understanding complex numbers and algebraic manipulation can be applied to simplify complex expressions.