Express `(1 /(2 -2i)+3/(1+i)) ((3+ 4i)/(2-4i))` in the form of a +ib

Answer

To express the given expression \((\frac{1}{2 - 2i} + \frac{3}{1 + i}) \cdot \frac{3 + 4i}{2 - 4i}\) in the form \(a + bi\), we will follow these steps:

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

\[ \frac{1}{2 - 2i} \cdot \frac{2 + 2i}{2 + 2i} = \frac{2 + 2i}{(2 - 2i)(2 + 2i)} = \frac{2 + 2i}{4 + 4} = \frac{2 + 2i}{8} = \frac{1}{4} + \frac{1}{4}i \]

### Step 2: Simplify \(\frac{3}{1 + i}\)
Similarly, we simplify \(\frac{3}{1 + i}\) by multiplying by the conjugate:

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

### Step 3: Combine the two fractions
Now we can combine the two simplified fractions:

\[ \frac{1}{4} + \frac{1}{4}i + \left(\frac{3}{2} - \frac{3}{2}i\right) \]

Finding a common denominator (4):

\[ \frac{1}{4} + \frac{3 \cdot 2}{4} - \left(\frac{3}{2}i + \frac{1}{4}i\right) = \frac{1 + 6}{4} - \left(\frac{6}{4}i + \frac{1}{4}i\right) = \frac{7}{4} - \frac{7}{4}i \]

### Step 4: Multiply by \(\frac{3 + 4i}{2 - 4i}\)
Next, we need to multiply this result by \(\frac{3 + 4i}{2 - 4i}\). First, simplify \(\frac{3 + 4i}{2 - 4i}\):

Multiply by the conjugate:

\[ \frac{3 + 4i}{2 - 4i} \cdot \frac{2 + 4i}{2 + 4i} = \frac{(3 + 4i)(2 + 4i)}{(2 - 4i)(2 + 4i)} \]

Calculating the denominator:

\[ (2 - 4i)(2 + 4i) = 4 + 16 = 20 \]

Calculating the numerator:

\[ (3 + 4i)(2 + 4i) = 6 + 12i + 8i + 16i^2 = 6 + 20i - 16 = -10 + 20i \]

So,

\[ \frac{3 + 4i}{2 - 4i} = \frac{-10 + 20i}{20} = -\frac{1}{2} + i \]

### Step 5: Multiply the results
Now we multiply \(\frac{7}{4} - \frac{7}{4}i\) by \(-\frac{1}{2} + i\):

\[ \left(\frac{7}{4} - \frac{7}{4}i\right) \cdot \left(-\frac{1}{2} + i\right) \]

Using the distributive property:

\[ = \frac{7}{4} \cdot -\frac{1}{2} + \frac{7}{4}i - \frac{7}{4}i \cdot -\frac{1}{2} - \frac{7}{4}i^2 \]

Calculating each term:

1. \(-\frac{7}{8}\)
2. \(+\frac{7}{4}i\)
3. \(+\frac{7}{8}i\)
4. \(+\frac{7}{4}\) (since \(i^2 = -1\))

Combining real and imaginary parts:

Real part: \(-\frac{7}{8} + \frac{7}{4} = -\frac{7}{8} + \frac{14}{8} = \frac{7}{8}\)

Imaginary part: \(\frac{7}{4}i + \frac{7}{8}i = \frac{14}{8}i + \frac{7}{8}i = \frac{21}{8}i\)

### Final Result
Thus, the expression in the form \(a + bi\) is:

\[ \frac{7}{8} + \frac{21}{8}i \]