Evaluate `{(1/(sqrt(27)))^(2-[(log_(5) 13)/(2log_(5) 9)])}^(1/2)`

Options

A. `(5sqrt(2))/(27)`
B. `(sqrt(2))/(27)`
C. `(4sqrt(2))/(27)`
D. `(2sqrt(2))/(27)`

Answer

To evaluate the expression \(\left(\frac{1}{\sqrt{27}}\right)^{2 - \frac{\log_{5} 13}{2 \log_{5} 9}}^{\frac{1}{2}}\), we will follow these steps:

### Step 1: Simplify the base
First, we simplify \(\frac{1}{\sqrt{27}}\):
\[ \sqrt{27} = \sqrt{3^3} = 3^{3/2} \]
Thus,
\[ \frac{1}{\sqrt{27}} = 3^{-3/2} \]

### Step 2: Simplify the exponent
Next, we focus on simplifying the exponent \(2 - \frac{\log_{5} 13}{2 \log_{5} 9}\). We can use properties of logarithms here.

Using the property \(\frac{\log_{a} b}{\log_{a} c} = \log_{c} b\):
\[ \frac{\log_{5} 13}{2 \log_{5} 9} = \frac{1}{2} \cdot \frac{\log_{5} 13}{\log_{5} 9} = \frac{1}{2} \log_{9} 13 \]
Thus, we can rewrite the exponent:
\[ 2 - \frac{\log_{5} 13}{2 \log_{5} 9} = 2 - \frac{1}{2} \log_{9} 13 \]

### Step 3: Combine the exponent
Now we can express the exponent as:
\[ 2 - \frac{1}{2} \log_{9} 13 = \frac{4}{2} - \frac{1}{2} \log_{9} 13 = \frac{4 - \log_{9} 13}{2} \]

### Step 4: Substitute back into the expression
Now substituting back into the expression:
\[ \left(3^{-3/2}\right)^{\frac{4 - \log_{9} 13}{2}} = 3^{-3/2 \cdot \frac{4 - \log_{9} 13}{2}} = 3^{-\frac{3(4 - \log_{9} 13)}{4}} \]

### Step 5: Final simplification
This simplifies to:
\[ 3^{-\frac{12 - 3 \log_{9} 13}{4}} = 3^{-\frac{12}{4} + \frac{3 \log_{9} 13}{4}} = 3^{-3 + \frac{3 \log_{9} 13}{4}} = \frac{3^{\frac{3 \log_{9} 13}{4}}}{27} \]

Using the property \(a^{\log_{a} b} = b\):
\[ 3^{\frac{3 \log_{9} 13}{4}} = 13^{\frac{3}{4}} \]
Thus, the final expression is:
\[ \frac{13^{\frac{3}{4}}}{27} \]

### Final Answer:
\[ \frac{13^{\frac{3}{4}}}{27} \]