The value of` ((log_(2)9)^(2))^(1/(log_(2)(log_(2)9)))xx(sqrt7)^(1/(log_(4)7))` is ________.

Explanation

To solve the expression \(((\log_{2}9)^{2})^{\frac{1}{\log_{2}(\log_{2}9)}} \times (\sqrt{7})^{\frac{1}{\log_{4}7}}\), we will break it down into two parts, \(a\) and \(b\), and then find the product \(e = a \times b\).

### Step 1: Simplifying Part \(a\)

Let:
\[ a = (\log_{2}9)^{2})^{\frac{1}{\log_{2}(\log_{2}9)}} \]

Using the property of logarithms:
\[ \frac{1}{\log_{b}a} = \log_{a}b \]
we can rewrite:
\[ a = (\log_{2}9)^{2})^{\log_{\log_{2}9}2} \]

Now, using the property \(x^{\log_{b}a} = a^{\log_{b}x}\), we can rewrite \(a\) as:
\[ a = 2^{\log_{\log_{2}9}(\log_{2}9)^{2}} \]

This simplifies to:
\[ a = 2^{2 \cdot \log_{\log_{2}9}(\log_{2}9)} = 2^{2} = 4 \]

### Step 2: Simplifying Part \(b\)

Let:
\[ b = (\sqrt{7})^{\frac{1}{\log_{4}7}} \]

We can rewrite \(\sqrt{7}\) as \(7^{1/2}\):
\[ b = (7^{1/2})^{\frac{1}{\log_{4}7}} = 7^{\frac{1}{2 \cdot \log_{4}7}} \]

Using the change of base formula again:
\[ \log_{4}7 = \frac{\log_{2}7}{\log_{2}4} = \frac{\log_{2}7}{2} \]
Thus, we have:
\[ b = 7^{\frac{1}{2 \cdot \frac{\log_{2}7}{2}}} = 7^{\frac{1}{\log_{2}7}} = 2^{\log_{7}7} = 2^{1} = 2 \]

### Step 3: Finding the Final Value

Now that we have both parts:
\[ a = 4 \quad \text{and} \quad b = 2 \]
We can find \(e\):
\[ e = a \times b = 4 \times 2 = 8 \]

Thus, the value of the expression is:
\[ \boxed{8} \]