We are tasked with solving the equation:

$\sqrt[5]{\log_3(x)} + \log_3\left(\sqrt{x}\right) = 12$

Step 1: Simplify $\log_3(\sqrt{x})$

We can rewrite $\log_3\left(\sqrt{x}\right)$ using logarithmic properties. Recall that $\log_b(a^n) = n \log_b(a)$, so:

$\log_3\left(\sqrt{x}\right) = \log_3\left(x^{1/2}\right) = \frac{1}{2} \log_3(x)$

Thus, the equation becomes:

$\sqrt[5]{\log_3(x)} + \frac{1}{2} \log_3(x) = 12$

Step 2: Let $y = \log_3(x)$

To simplify further, let's set $y = \log_3(x)$. The equation now becomes:

$\sqrt[5]{y} + \frac{1}{2}y = 12$

Step 3: Introduce a substitution for $\sqrt[5]{y}$

Let $z = \sqrt[5]{y}$, which means $z^5 = y$. Substituting this into the equation, we get:

[ z