Problem 76 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The result in standard form is \( 1.2 \).

Step 1: Simplify the denominators

To simplify the denominators, multiply both the numerator and denominator of each fraction by the conjugate of the denominator. This will result in a real number. The conjugate of a complex number is obtained by changing the sign in front of the imaginary part. Hence, \(\frac{1+i}{1+2 i}\) is multiplied by \(\frac{1-2 i}{1-2 i}\) and \(\frac{1-i}{1-2 i}\) is multiplied by \(\frac{1+2 i}{1+2 i}\).

Step 2: Calculate

That results in: \(\frac{(1+i)(1-2 i)}{(1+2 i)(1-2 i)} + \frac{(1-i)(1+2 i)}{(1-2 i)(1+2 i)}\). This simplifies to: \(\frac{1 - 2i + i - 2i^2}{1 - 4i^2} + \frac{1 + 2i - i - 2i^2}{1 - 4i^2}\).

Step 3: Simplify further

Recall that \(i^2 = -1\). Thus, \(\frac{1 - 2i + i + 2}{1 + 4} + \frac{1 + 2i - i + 2}{1 + 4}\),which simplifies to: \(\frac{3 - i}{5} + \frac{3 + i}{5}\).

Step 4: Final result

Finally, add the two fractions together: \(\frac{3 - i}{5} + \frac{3 + i}{5} = \frac{3+3}{5} = \frac{6}{5} = 1.2\)