The problem asks to find the value of the expression $((log_2 9)^2)^{\frac{1}{log_2(log_2 9)}} \times (\sqrt{7})^{\frac{1}{log_4 7}}$.

1. Problem Description

The problem asks to find the value of the expression $((log_2 9)^2)^{\frac{1}{log_2(log_2 9)}} \times (\sqrt{7})^{\frac{1}{log_4 7}}$.

2. Solution Steps

Let's analyze the first term.

$((log_2 9)^2)^{\frac{1}{log_2(log_2 9)}}$

We can rewrite this as:

$(log_2 9)^{\frac{2}{log_2(log_2 9)}}$

Using the property $a^{log_a x} = x$, we want to manipulate the exponent. We can rewrite the exponent as:

$\frac{2}{log_2(log_2 9)} = 2 * \frac{1}{log_2(log_2 9)} = 2 * log_{log_2 9} 2 = log_{log_2 9} 2^2 = log_{log_2 9} 4$

Thus, the first term is $(log_2 9)^{log_{log_2 9} 4} = 4$.

Now, let's analyze the second term:

$(\sqrt{7})^{\frac{1}{log_4 7}}$

We can rewrite this as:

$(7^{1/2})^{\frac{1}{log_4 7}} = 7^{\frac{1}{2} * \frac{1}{log_4 7}} = 7^{\frac{1}{2} * log_7 4} = 7^{log_7 4^{1/2}} = 7^{log_7 2} = 2$.

So, the expression is $4 \times 2 = 8$.

3. Final Answer

8