By what number should we multiply `(-6)^(-1)` to obtain a product equal to `9^(-1)` ?

Answer

To solve the problem of finding the number that should be multiplied by \((-6)^{-1}\) to obtain a product equal to \(9^{-1}\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find a number \(x\) such that:
\[ (-6)^{-1} \cdot x = 9^{-1} \]

2. **Rewrite the Expression**: We can express \((-6)^{-1}\) and \(9^{-1}\) in terms of their positive powers:
\[ (-6)^{-1} = \frac{1}{-6} \quad \text{and} \quad 9^{-1} = \frac{1}{9} \]

3. **Set Up the Equation**: Substitute these values into the equation:
\[ \frac{1}{-6} \cdot x = \frac{1}{9} \]

4. **Isolate \(x\)**: To find \(x\), we can multiply both sides of the equation by \(-6\):
\[ x = \frac{1}{9} \cdot (-6) \]

5. **Simplify the Expression**: Now, simplify the right-hand side:
\[ x = \frac{-6}{9} \]

6. **Reduce the Fraction**: We can simplify \(\frac{-6}{9}\) by dividing both the numerator and the denominator by 3:
\[ x = \frac{-2}{3} \]

### Final Answer:
Thus, the number we should multiply \((-6)^{-1}\) by to obtain \(9^{-1}\) is:
\[ \boxed{-\frac{2}{3}} \]