15. \( \sqrt[4]{8 \cdot \sqrt[4]{8 \cdot \sqrt[4]{8 \ldots}}}= \) ?

Let's denote \( x = \sqrt[4]{8 \cdot \sqrt[4]{8 \cdot \sqrt[4]{8 \ldots}}} \). This means that we can rewrite it as: \[ x = \sqrt[4]{8 \cdot x} \] Now, to eliminate the fourth root, we can raise both sides to the power of 4: \[ x^4 = 8 \cdot x \] Rearranging gives us: \[ x^4 - 8x = 0 \] Factoring out x: \[ x(x^3 - 8) = 0 \] This equation gives us two possibilities: \( x = 0 \) or \( x^3 - 8 = 0 \). Solving the second equation, we have: \[ x^3 = 8 \quad \Rightarrow \quad x = \sqrt[3]{8} = 2 \] So the solution is: \[ \sqrt[4]{8 \cdot \sqrt[4]{8 \cdot \sqrt[4]{8 \ldots}}} = 2 \] Therefore, the answer is: \[ \boxed{2} \]