Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The result of the operation in standard form is \(-0.5-2.5i\).

Step 1: Rationalize the Denominator of Each Fraction

To rationalize the denominator of each fraction, multiply the numerator and denominator of each fraction by the conjugate of the denominator. The conjugate of \(1+i\) is \(1-i\) and of \(1-i\) is \(1+i\). So we simplify both fractions like this: \[\frac{2}{1+i}*(1-i)/(1-i) - \frac{3}{1-i}*(1+i)/(1+i)\]

Step 2: Simplify Each Fraction

Next, perform the multiplications on top and bottom, remembering that \(i^2 = -1\). After evaluating the multiplications and simplifications, the expression turns as follows: \[\frac{2-2i}{2} - \frac{3+3i}{2}\]

Step 3: Simplify Further and Combine Like Terms

Now, divide each term within the fractions by the real number in the denominator. Combine the real parts together and the imaginary parts together to give the final expression in standard form: \[1-i - (1.5+1.5i)\] which simplifies to \[-0.5-2.5i\]