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Chapter 2: Problem 61

Perform the operation and write the result in standard form. $$\frac{2}{1+i}-\frac{3}{1-i}$$

Short Answer

Expert verified
The operation cannot be carried out because it involves division by zero.

Step by step solution

01

Rationalize the Denominators

First, multiply both the numerator and denominator of each fraction by the conjugate of the denominator. The conjugate of \(1+i\) is \(1-i\) and the conjugate of \(1-i\) is \(1+i\). So, we have \[\frac{2}{1+i}\cdot\frac{1-i}{1-i} - \frac{3}{1-i}\cdot\frac{1+i}{1+i} = \frac{2(1-i)}{1 - i^2} - \frac{3(1 + i)}{1 - (-i)^2}\].
02

Simplify Fractions

Simplify each expression using the fact that \(i^2 = -1\). Then, distribute the numerator of each fraction which gives us: \[\frac{2 - 2i}{1 - (-1)} - \frac{3 + 3i}{1 - 1} = \frac{2 - 2i}{2} - \frac{3 + 3i}{0}\].
03

Evaluate the Second Fraction

The second fraction becomes undefined because division by zero is undefined in mathematics. The operation cannot be carried out.

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