Compute the following `(81^(1/((log)_5 9))+3^(3/((log)_(sqrt(6))3)))/(409)dot((sqrt(7))^(2/((log)_(25)7))-(125)^((log)_(25)6))`

Options

A. 0
B. 1
C. 2
D. 3

Explanation

To solve the given expression

\[ \frac{81^{\frac{1}{\log_5 9}} + 3^{\frac{3}{\log_{\sqrt{6}} 3}}}{409 \cdot \left(\sqrt{7}^{\frac{2}{\log_{25} 7}} - 125^{\log_{25} 6}\right)} \]

we will apply logarithmic properties step by step.

### Step 1: Simplify \(81^{\frac{1}{\log_5 9}}\)

Using the property \(a^{\log_b c} = c^{\log_b a}\), we can rewrite \(81\) as \(9^2\):

\[ 81^{\frac{1}{\log_5 9}} = (9^2)^{\frac{1}{\log_5 9}} = 9^{\frac{2}{\log_5 9}} = 9^{\log_9 5^2} = 5^2 = 25 \]

**Hint:** Remember that \(a^{\log_b c} = c^{\log_b a}\) can help in simplifying expressions involving logarithms.

### Step 2: Simplify \(3^{\frac{3}{\log_{\sqrt{6}} 3}}\)

Using the same property, we rewrite \(\log_{\sqrt{6}} 3\):

\[ 3^{\frac{3}{\log_{\sqrt{6}} 3}} = 3^{\log_3 (\sqrt{6})^3} = (\sqrt{6})^3 = 6^{\frac{3}{2}} = 6\sqrt{6} \]

**Hint:** Converting logarithmic bases can often simplify the expression.

### Step 3: Combine the results from Steps 1 and 2

Now we can combine the results:

\[ 81^{\frac{1}{\log_5 9}} + 3^{\frac{3}{\log_{\sqrt{6}} 3}} = 25 + 6\sqrt{6} \]

**Hint:** Always combine like terms carefully and check for simplifications.

### Step 4: Simplify the denominator \(409 \cdot \left(\sqrt{7}^{\frac{2}{\log_{25} 7}} - 125^{\log_{25} 6}\right)\)

First, simplify \(\sqrt{7}^{\frac{2}{\log_{25} 7}}\):

\[ \sqrt{7}^{\frac{2}{\log_{25} 7}} = 7^{\frac{1}{\log_{25} 7}} = 7^{\log_7 25} = 25 \]

Now simplify \(125^{\log_{25} 6}\):

\[ 125 = 5^3 \implies 125^{\log_{25} 6} = (5^3)^{\log_{25} 6} = 6^{\log_{25} 125} = 6^{\frac{3}{2}} = 6\sqrt{6} \]

Now substituting back into the denominator:

\[ \sqrt{7}^{\frac{2}{\log_{25} 7}} - 125^{\log_{25} 6} = 25 - 6\sqrt{6} \]

**Hint:** When dealing with powers and logarithms, keep track of the bases and their relationships.

### Step 5: Substitute back into the full expression

Now we can substitute everything back into the expression:

\[ \frac{25 + 6\sqrt{6}}{409 \cdot (25 - 6\sqrt{6})} \]

### Step 6: Recognize the form of the expression

Notice that the numerator and denominator can be expressed in the form of \(a + b\) and \(a - b\):

\[ \frac{(25 + 6\sqrt{6})}{(25 - 6\sqrt{6})} = \frac{a + b}{a - b} \]

This can be simplified using the identity \(\frac{a + b}{a - b} = \frac{(a^2 - b^2)}{(a^2 - b^2)}\):

\[ = \frac{25^2 - (6\sqrt{6})^2}{409} \]

Calculating \(25^2 = 625\) and \((6\sqrt{6})^2 = 216\):

\[ = \frac{625 - 216}{409} = \frac{409}{409} = 1 \]

### Final Answer

Thus, the final answer is

\[ \boxed{1} \]