Which of the following complex numbers is equivalent to `(3+i)/(4-7i)`?

Options

A. `(17)/(28)`
B. `-(19)/(33)-(25)/(33)i`
C. `(1)/(13)-(5)/(13)i`
D. `(1)/(13)+(5)/(13)i`

Answer

To simplify the complex number \((3+i)/(4-7i)\), we follow these steps:

### Step 1: Identify the Conjugate of the Denominator
The denominator is \(4 - 7i\). The conjugate of a complex number \(a + bi\) is \(a - bi\). Thus, the conjugate of \(4 - 7i\) is \(4 + 7i\).

**Hint:** Remember that the conjugate of a complex number changes the sign of the imaginary part.

### Step 2: Multiply the Numerator and Denominator by the Conjugate
We multiply both the numerator and denominator by the conjugate of the denominator:
\[ \frac{3+i}{4-7i} \cdot \frac{4+7i}{4+7i} \]
This gives us:
\[ \frac{(3+i)(4+7i)}{(4-7i)(4+7i)} \]

**Hint:** Multiplying by the conjugate helps eliminate the imaginary part from the denominator.

### Step 3: Simplify the Numerator
Now we calculate the numerator:
\[ (3+i)(4+7i) = 3 \cdot 4 + 3 \cdot 7i + i \cdot 4 + i \cdot 7i \]
Calculating each term:
- \(3 \cdot 4 = 12\)
- \(3 \cdot 7i = 21i\)
- \(i \cdot 4 = 4i\)
- \(i \cdot 7i = 7i^2 = 7(-1) = -7\)

Combining these gives:
\[ 12 + 21i + 4i - 7 = (12 - 7) + (21i + 4i) = 5 + 25i \]

**Hint:** Combine like terms carefully, remembering that \(i^2 = -1\).

### Step 4: Simplify the Denominator
Next, we simplify the denominator:
\[ (4-7i)(4+7i) = 4^2 - (7i)^2 = 16 - 49(-1) = 16 + 49 = 65 \]

**Hint:** Use the difference of squares formula \(a^2 - b^2\) to simplify the denominator.

### Step 5: Combine the Results
Now we have:
\[ \frac{5 + 25i}{65} \]
We can separate this into real and imaginary parts:
\[ \frac{5}{65} + \frac{25i}{65} = \frac{1}{13} + \frac{5}{13}i \]

**Hint:** Simplifying fractions can help in identifying the final form of the complex number.

### Final Answer
The simplified form of the complex number \((3+i)/(4-7i)\) is:
\[ \frac{1}{13} + \frac{5}{13}i \]