Given that $A + B + C = 180^\circ$ (since $A$, $B$, and $C$ are angles in a triangle), we can use the property that:

$\tan(A) \cdot \tan(B) \cdot \tan(C) = \tan(A + B + C) = \tan(180^\circ) = 0$

However, in this problem, the expression under the square root simplifies as follows:

We are tasked with finding the value of:

$\sqrt{\frac{\tan^8 A + \tan^8 B + \tan^8 C}{\tan A \cdot \tan B \cdot \tan C}}$

Since $\tan(A) \cdot \tan(B) \cdot \tan(C) = 0$, the denominator of this expression becomes zero, which makes the entire expression undefined.

Therefore, the value of the given expression is undefined for any triangle where the sum of the angles equals $180^\circ$.

Would you like me to explain further or try a different approach? Here are 5 related questions:

1. How do we find the tangent of angles in a triangle?
2. What are the properties of the sum of angles in a triangle?
3. What happens when the tangent product of angles in a triangle is zero?
4. How can we simplify trigonometric expressions in terms of a triangle's angles?
5. Are there special triangles where tangent functions have specific values?

Tip: Always check if trigonometric identities simplify an expression before calculating it directly.