If `(2rs)^(-1)=3s^(-2)`, what is the value of `(r)/(s)`?

Answer

To solve the equation \( (2rs)^{-1} = 3s^{-2} \), we can follow these steps:

### Step 1: Rewrite the equation
The expression \( (2rs)^{-1} \) can be rewritten as:
\[ \frac{1}{2rs} = 3s^{-2} \]

### Step 2: Rewrite \( s^{-2} \)
The term \( s^{-2} \) can be rewritten as:
\[ s^{-2} = \frac{1}{s^2} \]
Thus, we can rewrite the right side of the equation:
\[ 3s^{-2} = \frac{3}{s^2} \]

### Step 3: Set the equation
Now we have:
\[ \frac{1}{2rs} = \frac{3}{s^2} \]

### Step 4: Cross-multiply
Cross-multiplying gives us:
\[ 1 \cdot s^2 = 3 \cdot 2rs \]
This simplifies to:
\[ s^2 = 6rs \]

### Step 5: Rearranging the equation
To isolate \( r \), we can rearrange the equation:
\[ s^2 = 6rs \implies \frac{s^2}{s} = 6r \implies s = 6r \]

### Step 6: Solve for \( \frac{r}{s} \)
Now, we can express \( r \) in terms of \( s \):
\[ r = \frac{s}{6} \]
To find \( \frac{r}{s} \):
\[ \frac{r}{s} = \frac{\frac{s}{6}}{s} = \frac{1}{6} \]

### Final Answer
Thus, the value of \( \frac{r}{s} \) is:
\[ \frac{r}{s} = \frac{1}{6} \]

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