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

Answer

To express the given expression \(\left(\frac{1}{2 - 2i} + \frac{3}{1 + i}\right) \left(\frac{3 + 4i}{2 - 4i}\right)\) in the form \(a + ib\), we will simplify each part step by step.

### 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:

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

Calculating the denominator:

\[ (2 - 2i)(2 + 2i) = 2^2 - (2i)^2 = 4 - 4(-1) = 4 + 4 = 8 \]

Thus, we have:

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

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

Next, we simplify \(\frac{3}{1 + i}\) by multiplying by the conjugate:

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

Calculating the denominator:

\[ (1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]

Thus, we have:

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

### Step 3: Combine the two fractions

Now we 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 (which is 4):

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

This simplifies to:

\[ \frac{7}{4} - \frac{7}{4}i \]

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

Now we simplify \(\frac{3 + 4i}{2 - 4i}\) by multiplying 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:

\[ (2 - 4i)(2 + 4i) = 2^2 - (4i)^2 = 4 - 16(-1) = 4 + 16 = 20 \]

Calculating the numerator:

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

Thus, we have:

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

### Step 5: Multiply the two results

Now we multiply the results from Step 3 and Step 4:

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

Using the distributive property:

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

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 the real and imaginary parts:

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

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

Thus, the final result is:

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

### Final Answer:

The expression \(\left(\frac{1}{2 - 2i} + \frac{3}{1 + i}\right) \left(\frac{3 + 4i}{2 - 4i}\right)\) in the form \(a + ib\) is:

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