The value of $\left((\log_2 9)^2\right)^{\frac{1}{\log_2(\log_2 9)}} \times (\sqrt{7})^{\frac{1}{\log_4 7}}$ is ________.

The value of ((log₂ 9)²)^(1/(log₂(log₂ 9))) × (√7)^(1/(log₄ 7)) is ________.

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

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

$\text{To find: value of the whole expression}$

$\log_b a = \dfrac{1}{\log_a b}$ $b^{\log_b x} = x$ $1.\ \text{simplify factor }1$ $2.\ \text{simplify factor }2$ $3.\ \text{multiply}$ $a = \log_2 9$ $F_1 = (a^2)^{\frac{1}{\log_2 a}} = a^{\frac{2}{\log_2 a}}$ $\dfrac{2}{\log_2 a} = 2\log_a 2 = \log_a 4$ $F_1 = a^{\log_a 4} = 4$ $F_2 = (\sqrt7)^{\frac{1}{\log_4 7}}$ $\dfrac{1}{\log_4 7} = \log_7 4$ $F_2 = (\sqrt7)^{\log_7 4}$ $F_2 = \left(7^{1/2}\right)^{\log_7 4} = 7^{\frac12\log_7 4} = 4^{1/2} = 2$ $F_1 \times F_2 = 4 \times 2 = 8$ $\boxed{8}$