Find the value of x satisfying the equations `log_(3)(log_(2)x)+log_(1//3)(log_(1//2)y)=1` and `xy^(2)=9`

Explanation

To solve the equations \( \log_{3}(\log_{2} x) + \log_{\frac{1}{3}}(\log_{\frac{1}{2}} y) = 1 \) and \( xy^{2} = 9 \), we will follow these steps:

### Step 1: Rewrite the second logarithm
The equation can be rewritten using the change of base formula. The logarithm \( \log_{\frac{1}{3}}(\log_{\frac{1}{2}} y) \) can be expressed as:
\[ \log_{\frac{1}{3}}(\log_{\frac{1}{2}} y) = -\log_{3}(\log_{\frac{1}{2}} y) \]
Thus, the original equation becomes:
\[ \log_{3}(\log_{2} x) - \log_{3}(\log_{\frac{1}{2}} y) = 1 \]

### Step 2: Combine the logarithms
Using the properties of logarithms, we can combine the two logarithmic terms:
\[ \log_{3}\left(\frac{\log_{2} x}{\log_{\frac{1}{2}} y}\right) = 1 \]

### Step 3: Exponentiate to eliminate the logarithm
Exponentiating both sides gives:
\[ \frac{\log_{2} x}{\log_{\frac{1}{2}} y} = 3 \]

### Step 4: Rewrite \( \log_{\frac{1}{2}} y \)
We know that \( \log_{\frac{1}{2}} y = -\log_{2} y \). Substituting this into the equation gives:
\[ \frac{\log_{2} x}{-\log_{2} y} = 3 \]
This can be rearranged to:
\[ \log_{2} x = -3 \log_{2} y \]

### Step 5: Rewrite in exponential form
This implies:
\[ \log_{2} x = \log_{2} y^{-3} \]
Thus, we can equate the arguments:
\[ x = \frac{1}{y^{3}} \]

### Step 6: Substitute into the second equation
Now, substitute \( x \) into the second equation \( xy^{2} = 9 \):
\[ \left(\frac{1}{y^{3}}\right)y^{2} = 9 \]
This simplifies to:
\[ \frac{y^{2}}{y^{3}} = 9 \quad \Rightarrow \quad \frac{1}{y} = 9 \quad \Rightarrow \quad y = \frac{1}{9} \]

### Step 7: Find \( x \)
Now that we have \( y \), substitute back to find \( x \):
\[ x = \frac{1}{y^{3}} = \frac{1}{\left(\frac{1}{9}\right)^{3}} = \frac{1}{\frac{1}{729}} = 729 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{729} \]