` ( 8 - i)/( 3 - 2i ) `
If the expression above is rewritten in the form ` a + bi`, where ` a and b ` are real numbers, what is the value of ` a ` ? ( Note : ` i=sqrt ( -1) `)

Options

A. 2
B. ` ( 8)/(3) `
C. ` 3 `
D. ` (11 ) /(3) `

Explanation

To rewrite the expression \( \frac{8 - i}{3 - 2i} \) in the form \( a + bi \), where \( a \) and \( b \) are real numbers, we will follow these steps:

### Step 1: Multiply by the Conjugate
To eliminate the imaginary part in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 3 - 2i \) is \( 3 + 2i \).

\[ \frac{8 - i}{3 - 2i} \cdot \frac{3 + 2i}{3 + 2i} \]

### Step 2: Expand the Numerator
Now, we will expand the numerator:

\[ (8 - i)(3 + 2i) = 8 \cdot 3 + 8 \cdot 2i - i \cdot 3 - i \cdot 2i \]

Calculating each term:

\[ = 24 + 16i - 3i - 2i^2 \]

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

\[ = 24 + 16i - 3i + 2 = 26 + 13i \]

### Step 3: Expand the Denominator
Now, we will expand the denominator:

\[ (3 - 2i)(3 + 2i) = 3^2 - (2i)^2 = 9 - 4(-1) = 9 + 4 = 13 \]

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

\[ \frac{26 + 13i}{13} \]

### Step 5: Simplify
We can simplify this expression by dividing each term in the numerator by the denominator:

\[ = \frac{26}{13} + \frac{13i}{13} = 2 + i \]

### Conclusion
Now, we have the expression in the form \( a + bi \) where \( a = 2 \) and \( b = 1 \). Therefore, the value of \( a \) is:

\[ \boxed{2} \]