Perform the indicated operation and find the result in the form `a+i b :(3-sqrt(-16))/(1-sqrt(-9))`

Explanation

To solve the expression \((3 - \sqrt{-16}) / (1 - \sqrt{-9})\) and express it in the form \(a + ib\), we will follow these steps:

### Step 1: Simplify the square roots of negative numbers
We know that \(\sqrt{-1} = i\). Therefore, we can rewrite the square roots:
- \(\sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i\)
- \(\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i\)

Now, substitute these values into the expression:
\[ \frac{3 - \sqrt{-16}}{1 - \sqrt{-9}} = \frac{3 - 4i}{1 - 3i} \]

### Step 2: Rationalize the denominator
To eliminate the imaginary part in the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{3 - 4i}{1 - 3i} \cdot \frac{1 + 3i}{1 + 3i} \]

### Step 3: Multiply the numerators and denominators
Now, we will multiply the numerators and denominators:
- **Numerator**:
\[ (3 - 4i)(1 + 3i) = 3 \cdot 1 + 3 \cdot 3i - 4i \cdot 1 - 4i \cdot 3i = 3 + 9i - 4i - 12(-1) = 3 + 5i + 12 = 15 + 5i \]

- **Denominator**:
\[ (1 - 3i)(1 + 3i) = 1^2 - (3i)^2 = 1 - 9(-1) = 1 + 9 = 10 \]

### Step 4: Combine the results
Now we can write the expression as:
\[ \frac{15 + 5i}{10} \]

### Step 5: Simplify the fraction
We can separate the real and imaginary parts:
\[ \frac{15}{10} + \frac{5i}{10} = \frac{3}{2} + \frac{1}{2}i \]

### Final Result
Thus, the result in the form \(a + ib\) is:
\[ \frac{3}{2} + \frac{1}{2}i \]