Example 7 Multiplicative Magic Square
Figure 1 shows a partially filled magic square. Fill in the following nine numbers: $\frac{1}{4}, \frac{1}{2}, 1,2,4,8,16,32,64$ in the grid so that the product of the numbers in each row, column, and diagonal is the same. The number that should be filled in the “ $x$ ” cell is $\qquad$ .
Explanation: Label the unfilled cells with letters, as shown in Figure 2.
The product of the nine known numbers is
$$
\frac{1}{4} \times \frac{1}{2} \times 8 \times 1 \times 2 \times 32 \times 4 \times 16 \times 64=64^{3} \text {. }
$$

Therefore, the fixed product of the three numbers in each row, each column, and each diagonal is 64.

Since the fixed product is 64, we get from the second column and the third row respectively
$$
a c=1, e f=1 .
$$

Thus, $a, c, e, f$ are each one of the numbers $\frac{1}{4}, \frac{1}{2}, 2, 4$.
Considering the product of the first row, we get $a x=2$.
This means $x$ can only be 1 or 8.
Considering the product of the diagonal, we get $64=c e x$.
If $x=1$, then $c e=64$, which is impossible.
The only possibility is $x=8$. The filling method is shown in Figure 3.
The answer to this problem is: The number to be filled in the “$x$” cell should be 8.