Perform the indicated operation and give your answer in the form `x+yi`, where x and y are real numbers and `i= sqrt(-1)`
`(2- sqrt(-25))/(1- sqrt(-16))`

Answer

To solve the expression \((2 - \sqrt{-25}) / (1 - \sqrt{-16})\) and express it in the form \(x + yi\), we will follow these steps:

### Step 1: Simplify the square roots of negative numbers
We start by rewriting the square roots of the negative numbers using \(i\), where \(i = \sqrt{-1}\).

\[ \sqrt{-25} = \sqrt{-1 \cdot 25} = \sqrt{-1} \cdot \sqrt{25} = 5i \]
\[ \sqrt{-16} = \sqrt{-1 \cdot 16} = \sqrt{-1} \cdot \sqrt{16} = 4i \]

### Step 2: Substitute back into the expression
Now we substitute these values back into the original expression:

\[ \frac{2 - \sqrt{-25}}{1 - \sqrt{-16}} = \frac{2 - 5i}{1 - 4i} \]

### Step 3: 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, which is \(1 + 4i\):

\[ \frac{(2 - 5i)(1 + 4i)}{(1 - 4i)(1 + 4i)} \]

### Step 4: Expand the numerator and denominator
Now we expand both the numerator and the denominator:

**Numerator:**
\[ (2 - 5i)(1 + 4i) = 2 \cdot 1 + 2 \cdot 4i - 5i \cdot 1 - 5i \cdot 4i \]
\[ = 2 + 8i - 5i - 20i^2 \]
Since \(i^2 = -1\), we have:
\[ = 2 + 3i + 20 = 22 + 3i \]

**Denominator:**
\[ (1 - 4i)(1 + 4i) = 1^2 - (4i)^2 = 1 - 16i^2 \]
Again, since \(i^2 = -1\):
\[ = 1 + 16 = 17 \]

### Step 5: Combine the results
Now we can combine the results from the numerator and the denominator:

\[ \frac{22 + 3i}{17} \]

### Step 6: Separate into real and imaginary parts
We can express this as:

\[ \frac{22}{17} + \frac{3}{17}i \]

### Final Answer
Thus, the expression in the form \(x + yi\) is:

\[ \frac{22}{17} + \frac{3}{17}i \]

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