`(2/(1-i) + 3/(1+i))((2+3i)/(4+5i))`is equal to

Options

A. ` -117/82 - 13/82 i`
B. `-117/82 + 13/82 i`
C. `117/82 - (13i)/82`
D. `117/82 + ( 13i)/82 `

Answer

To solve the expression \((\frac{2}{1-i} + \frac{3}{1+i})\left(\frac{2+3i}{4+5i}\right)\), we will follow these steps:

### Step 1: Simplify \(\frac{2}{1-i}\) and \(\frac{3}{1+i}\)

To simplify \(\frac{2}{1-i}\), we multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{2}{1-i} \cdot \frac{1+i}{1+i} = \frac{2(1+i)}{(1-i)(1+i)} = \frac{2(1+i)}{1^2 - i^2} = \frac{2(1+i)}{1 - (-1)} = \frac{2(1+i)}{2} = 1+i \]

Now, for \(\frac{3}{1+i}\):

\[ \frac{3}{1+i} \cdot \frac{1-i}{1-i} = \frac{3(1-i)}{(1+i)(1-i)} = \frac{3(1-i)}{1^2 - i^2} = \frac{3(1-i)}{1 - (-1)} = \frac{3(1-i)}{2} = \frac{3}{2} - \frac{3}{2}i \]

### Step 2: Combine the two fractions

Now, we can combine \(1+i\) and \(\frac{3}{2} - \frac{3}{2}i\):

\[ 1+i + \left(\frac{3}{2} - \frac{3}{2}i\right) = \left(1 + \frac{3}{2}\right) + \left(1 - \frac{3}{2}\right)i = \frac{5}{2} - \frac{1}{2}i \]

### Step 3: Simplify \(\frac{2+3i}{4+5i}\)

Next, we simplify \(\frac{2+3i}{4+5i}\) by multiplying by the conjugate of the denominator:

\[ \frac{2+3i}{4+5i} \cdot \frac{4-5i}{4-5i} = \frac{(2+3i)(4-5i)}{(4+5i)(4-5i)} = \frac{8 - 10i + 12i - 15i^2}{16 - 25(-1)} = \frac{8 + 2i + 15}{41} = \frac{23 + 2i}{41} \]

### Step 4: Multiply the two results

Now we multiply \(\left(\frac{5}{2} - \frac{1}{2}i\right)\) and \(\left(\frac{23 + 2i}{41}\right)\):

\[ \left(\frac{5}{2} - \frac{1}{2}i\right) \cdot \left(\frac{23 + 2i}{41}\right) = \frac{1}{41} \left( \left(\frac{5}{2} \cdot 23 + \frac{5}{2} \cdot 2i - \frac{1}{2}i \cdot 23 - \frac{1}{2}i \cdot 2i\right) \right) \]

Calculating each term:

- \(\frac{5}{2} \cdot 23 = \frac{115}{2}\)
- \(\frac{5}{2} \cdot 2i = 5i\)
- \(-\frac{1}{2}i \cdot 23 = -\frac{23}{2}i\)
- \(-\frac{1}{2}i \cdot 2i = 1\)

Combining these:

\[ \frac{1}{41} \left( \frac{115}{2} + 1 + (5 - \frac{23}{2})i \right) = \frac{1}{41} \left( \frac{117}{2} + \left(5 - \frac{23}{2}\right)i \right) \]

Calculating \(5 - \frac{23}{2} = \frac{10}{2} - \frac{23}{2} = -\frac{13}{2}\):

\[ = \frac{1}{41} \left( \frac{117}{2} - \frac{13}{2}i \right) = \frac{117}{82} - \frac{13}{82}i \]

### Final Result

Thus, the final result is:

\[ \frac{117}{82} - \frac{13}{82}i \]

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