Simplify:
`-sqrt((-7)/(4))-sqrt((-1)/(7))`

Answer

To simplify the expression \(-\sqrt{\left(-\frac{7}{4}\right)} - \sqrt{\left(-\frac{1}{7}\right)}\), we can follow these steps:

### Step 1: Rewrite the square roots
We know that \(\sqrt{-x} = i\sqrt{x}\), where \(i\) is the imaginary unit. Therefore, we can rewrite the expression as:
\[ -\sqrt{\left(-\frac{7}{4}\right)} - \sqrt{\left(-\frac{1}{7}\right)} = -i\sqrt{\frac{7}{4}} - i\sqrt{\frac{1}{7}} \]

### Step 2: Simplify the square roots
Now, we simplify the square roots:
\[ -i\sqrt{\frac{7}{4}} = -i\frac{\sqrt{7}}{2} \quad \text{and} \quad -i\sqrt{\frac{1}{7}} = -i\frac{1}{\sqrt{7}} \]
Thus, we have:
\[ -i\frac{\sqrt{7}}{2} - i\frac{1}{\sqrt{7}} \]

### Step 3: Combine the terms
Now we can factor out \(-i\):
\[ -i\left(\frac{\sqrt{7}}{2} + \frac{1}{\sqrt{7}}\right) \]

### Step 4: Find a common denominator
To combine the terms inside the parentheses, we need a common denominator. The common denominator for \(2\) and \(\sqrt{7}\) is \(2\sqrt{7}\):
\[ \frac{\sqrt{7}}{2} = \frac{\sqrt{7} \cdot \sqrt{7}}{2\sqrt{7}} = \frac{7}{2\sqrt{7}} \]
\[ \frac{1}{\sqrt{7}} = \frac{2}{2\sqrt{7}} \]
Now we can add these fractions:
\[ \frac{7}{2\sqrt{7}} + \frac{2}{2\sqrt{7}} = \frac{7 + 2}{2\sqrt{7}} = \frac{9}{2\sqrt{7}} \]

### Step 5: Substitute back
Now substitute this back into our expression:
\[ -i\left(\frac{9}{2\sqrt{7}}\right) \]
This simplifies to:
\[ -\frac{9i}{2\sqrt{7}} \]

### Step 6: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and denominator by \(\sqrt{7}\):
\[ -\frac{9i\sqrt{7}}{2 \cdot 7} = -\frac{9i\sqrt{7}}{14} \]

### Final Answer
Thus, the simplified form of the expression is:
\[ -\frac{9i\sqrt{7}}{14} \]

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