Find the value of`log_(1//3)root(4)(729*root(3)(9^(-1)*27^(-4//3)))`.

Explanation

To solve the problem, we will break it down step by step.

### Step 1: Rewrite the expression
We start with the expression:
\[ \log_{(1/3)} \left( \sqrt[4]{729} \cdot \sqrt[3]{9^{-1} \cdot 27^{-4/3}} \right) \]

### Step 2: Simplify \( \sqrt[4]{729} \)
First, we simplify \( \sqrt[4]{729} \):
\[ 729 = 3^6 \quad \text{(since \( 729 = 27^2 = (3^3)^2 = 3^6 \))} \]
Thus,
\[ \sqrt[4]{729} = (3^6)^{1/4} = 3^{6/4} = 3^{3/2} \]

### Step 3: Simplify \( \sqrt[3]{9^{-1} \cdot 27^{-4/3}} \)
Next, we simplify \( \sqrt[3]{9^{-1} \cdot 27^{-4/3}} \):
\[ 9 = 3^2 \quad \text{and} \quad 27 = 3^3 \]
So,
\[ 9^{-1} = (3^2)^{-1} = 3^{-2} \quad \text{and} \quad 27^{-4/3} = (3^3)^{-4/3} = 3^{-4} \]
Now combine these:
\[ 9^{-1} \cdot 27^{-4/3} = 3^{-2} \cdot 3^{-4} = 3^{-6} \]
Now, we take the cube root:
\[ \sqrt[3]{3^{-6}} = (3^{-6})^{1/3} = 3^{-6/3} = 3^{-2} \]

### Step 4: Combine the results
Now we combine the results from Steps 2 and 3:
\[ \sqrt[4]{729} \cdot \sqrt[3]{9^{-1} \cdot 27^{-4/3}} = 3^{3/2} \cdot 3^{-2} = 3^{3/2 - 2} = 3^{3/2 - 4/2} = 3^{-1/2} \]

### Step 5: Substitute back into the logarithm
Now we substitute back into the logarithm:
\[ \log_{(1/3)}(3^{-1/2}) \]

### Step 6: Use the change of base formula
Using the property of logarithms:
\[ \log_{a}(b^c) = c \cdot \log_{a}(b) \]
we have:
\[ \log_{(1/3)}(3^{-1/2}) = -\frac{1}{2} \cdot \log_{(1/3)}(3) \]

### Step 7: Evaluate \( \log_{(1/3)}(3) \)
Using the change of base formula:
\[ \log_{(1/3)}(3) = \frac{1}{\log_{3}(1/3)} = \frac{1}{-1} = -1 \]
Thus,
\[ \log_{(1/3)}(3) = -1 \]

### Step 8: Final calculation
Now substituting back:
\[ -\frac{1}{2} \cdot (-1) = \frac{1}{2} \]

### Final Answer
The value of the expression is:
\[ \frac{1}{2} \]

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