The problem asks us to solve for $x$ in the logarithmic equation $\log_{64}(x) = -\frac{1}{2}$ by converting the equation to exponential form.

1. Problem Description

The problem asks us to solve for $x$ in the logarithmic equation $\log_{64}(x) = -\frac{1}{2}$ by converting the equation to exponential form.

2. Solution Steps

To solve for $x$, we need to rewrite the logarithmic equation in exponential form. The general relationship between logarithms and exponentials is given by:

$\log_b(a) = c$ is equivalent to $b^c = a$.

In our case, $b = 64$, $a = x$, and $c = -\frac{1}{2}$.

Therefore, the exponential form of the equation is:

$64^{-\frac{1}{2}} = x$

Since $a^{-n} = \frac{1}{a^n}$, we can rewrite the left side as:

$x = \frac{1}{64^{\frac{1}{2}}}$

We know that $a^{\frac{1}{2}} = \sqrt{a}$, so $64^{\frac{1}{2}} = \sqrt{64} = 8$.

Therefore,

$x = \frac{1}{8}$.

3. Final Answer

The final answer is $\frac{1}{8}$.