Simiplify : `sqrt(-(49)/(25)) sqrt(-(1)/(9))`

Answer

To simplify the expression \( \sqrt{-\frac{49}{25}} \cdot \sqrt{-\frac{1}{9}} \), we can follow these steps:

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

\[ \sqrt{-\frac{49}{25}} = \sqrt{-1} \cdot \sqrt{\frac{49}{25}} = i \cdot \frac{7}{5} \]

\[ \sqrt{-\frac{1}{9}} = \sqrt{-1} \cdot \sqrt{\frac{1}{9}} = i \cdot \frac{1}{3} \]

### Step 2: Multiply the results
Now, we can multiply the two results together:

\[ \sqrt{-\frac{49}{25}} \cdot \sqrt{-\frac{1}{9}} = \left(i \cdot \frac{7}{5}\right) \cdot \left(i \cdot \frac{1}{3}\right) \]

### Step 3: Combine the terms
When we multiply these, we can combine the coefficients and the \( i \) terms:

\[ = i \cdot i \cdot \frac{7}{5} \cdot \frac{1}{3} \]

### Step 4: Simplify the coefficients
Recall that \( i \cdot i = i^2 = -1 \). Therefore, we can simplify:

\[ = -1 \cdot \left(\frac{7}{5} \cdot \frac{1}{3}\right) \]

### Step 5: Calculate the product of the fractions
Now, we calculate the product of the fractions:

\[ \frac{7}{5} \cdot \frac{1}{3} = \frac{7 \cdot 1}{5 \cdot 3} = \frac{7}{15} \]

### Step 6: Final result
Putting it all together, we have:

\[ = -\frac{7}{15} \]

Thus, the simplified expression is:

\[ \boxed{-\frac{7}{15}} \]