`(7 +i)/(8-i)`
If the expression above is expressed in the form `a +bi,` where `I = sqrt(-1,` what is the value of b ?

Options

A. `-1`
B. `3/13`
C. `11/13`
D. `15/64`

Answer

To solve the expression \((7 + i)/(8 - i)\) and express it in the form \(a + bi\), we will follow these steps:

### Step 1: Rationalize the Denominator
To eliminate the imaginary part in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(8 + i\).

\[ \frac{7 + i}{8 - i} \cdot \frac{8 + i}{8 + i} \]

### Step 2: Multiply the Numerators
Now, we will multiply the numerators:

\[ (7 + i)(8 + i) = 7 \cdot 8 + 7 \cdot i + i \cdot 8 + i \cdot i \]

Calculating this gives:

\[ 56 + 7i + 8i + i^2 \]

Since \(i^2 = -1\), we can substitute that in:

\[ 56 + 15i - 1 = 55 + 15i \]

### Step 3: Multiply the Denominators
Next, we multiply the denominators:

\[ (8 - i)(8 + i) = 8^2 - i^2 = 64 - (-1) = 64 + 1 = 65 \]

### Step 4: Combine the Results
Now we can combine the results from the numerator and denominator:

\[ \frac{55 + 15i}{65} \]

### Step 5: Separate the Real and Imaginary Parts
We can separate this into real and imaginary parts:

\[ \frac{55}{65} + \frac{15}{65}i \]

### Step 6: Simplify the Fractions
Now, simplify the fractions:

\[ \frac{55}{65} = \frac{11}{13} \quad \text{and} \quad \frac{15}{65} = \frac{3}{13} \]

Thus, we have:

\[ \frac{11}{13} + \frac{3}{13}i \]

### Step 7: Identify \(b\)
In the form \(a + bi\), we can see that \(b = \frac{3}{13}\).

### Final Answer
The value of \(b\) is \(\frac{3}{13}\).

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