If `a gt 0 ,b gt 0` then `sqrt(-a) * sqrtb` is equal to

Options

A. `-sqrt(ab)`
B. `sqrt(ab) i`
C. `sqrt(ab)`
D. none of these

Answer

To solve the expression \( \sqrt{-a} \cdot \sqrt{b} \) given that \( a > 0 \) and \( b > 0 \), we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the square root of a negative number**:
We know that \( \sqrt{-x} = \sqrt{-1} \cdot \sqrt{x} \). In this case, we can apply this to \( \sqrt{-a} \):
\[ \sqrt{-a} = \sqrt{-1} \cdot \sqrt{a} \]

2. **Substitute the value of \( \sqrt{-1} \)**:
The value of \( \sqrt{-1} \) is represented by the imaginary unit \( i \). Therefore, we can rewrite the expression as:
\[ \sqrt{-a} = i \cdot \sqrt{a} \]

3. **Substitute \( \sqrt{-a} \) back into the original expression**:
Now we substitute \( \sqrt{-a} \) into the expression \( \sqrt{-a} \cdot \sqrt{b} \):
\[ \sqrt{-a} \cdot \sqrt{b} = (i \cdot \sqrt{a}) \cdot \sqrt{b} \]

4. **Combine the square roots**:
We can combine the square roots since they are both under the same multiplication:
\[ \sqrt{-a} \cdot \sqrt{b} = i \cdot \sqrt{a} \cdot \sqrt{b} = i \cdot \sqrt{a \cdot b} \]

5. **Final answer**:
Thus, the expression \( \sqrt{-a} \cdot \sqrt{b} \) simplifies to:
\[ \sqrt{-a} \cdot \sqrt{b} = i \cdot \sqrt{a \cdot b} \]

### Final Result:
\[ \sqrt{-a} \cdot \sqrt{b} = i \cdot \sqrt{a \cdot b} \]