`(3-sqrt(-16))/(1-sqrt(-25))` is equal to

Options

A. `(-1)/24`
B. 0
C. `23/26 + 11/26 i`
D. 23+5i

Answer

To solve the expression \((3 - \sqrt{-16}) / (1 - \sqrt{-25})\), we will follow these steps:

### Step 1: Simplify the square roots of negative numbers
We know that \(\sqrt{-1} = i\), where \(i\) is the imaginary unit. Therefore, we can rewrite the square roots:
\[ \sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i \]
\[ \sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i \]
Now, substitute these values back into the expression:
\[ \frac{3 - 4i}{1 - 5i} \]

### Step 2: Multiply by the complex conjugate
To eliminate the imaginary part from the denominator, we multiply the numerator and the denominator by the complex conjugate of the denominator, which is \(1 + 5i\):
\[ \frac{(3 - 4i)(1 + 5i)}{(1 - 5i)(1 + 5i)} \]

### Step 3: Simplify the denominator
The denominator can be simplified using the difference of squares:
\[ (1 - 5i)(1 + 5i) = 1^2 - (5i)^2 = 1 - 25(-1) = 1 + 25 = 26 \]

### Step 4: Expand the numerator
Now, we need to expand the numerator:
\[ (3 - 4i)(1 + 5i) = 3 \cdot 1 + 3 \cdot 5i - 4i \cdot 1 - 4i \cdot 5i \]
Calculating each term:
\[ = 3 + 15i - 4i - 20i^2 \]
Since \(i^2 = -1\), we have:
\[ -20i^2 = 20 \]
Thus, combining the terms:
\[ = 3 + 15i - 4i + 20 = 23 + 11i \]

### Step 5: Combine the results
Now, we can write the entire expression:
\[ \frac{23 + 11i}{26} \]

### Final Result
This can be separated into real and imaginary parts:
\[ = \frac{23}{26} + \frac{11}{26}i \]

### Conclusion
Thus, the final answer is:
\[ \frac{23}{26} + \frac{11}{26}i \]

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