Given:
\(x \cdot \frac{1}{x} = 4\)

To Find:
The value of \(x'^4 \cdot \frac{1}{{x'^4}}\)

Solution:
We can simplify the given equation and then solve for \(x\) to find its value. Once we have the value of \(x\), we can substitute it into the expression to evaluate the final result.

Simplifying the equation:
\(x \cdot \frac{1}{x} = 4\)

The product of any number and its reciprocal is always equal to 1:
\(x \cdot \frac{1}{x} = 1\)

Therefore, we can rewrite the equation as:
\(1 = 4\)

This equation is not possible as it contradicts the basic properties of numbers. Hence, there is no solution to the given equation.

Conclusion:
Since the equation \(x \cdot \frac{1}{x} = 4\) has no solution, we cannot determine the value of \(x'^4 \cdot \frac{1}{{x'^4}}\) as it involves the unknown value of \(x\).