To solve the equation
\[ \frac{4x^{7}}{6} + 16 = 19131892, \]
we can start by isolating the term involving \( x \). First, subtract 16 from both sides:
\[ \frac{4x^{7}}{6} = 19131892 - 16, \]
which simplifies to:
\[ \frac{4x^{7}}{6} = 19131876. \]
Next, multiply both sides by \( \frac{6}{4} \) (or \( 1.5 \)) to isolate \( x^7 \):
\[ x^{7} = 19131876 \cdot \frac{6}{4}. \]
Calculating the right side:
\[ x^{7} = 19131876 \cdot 1.5 = 19131876 \cdot 1.5 = 19131876 \cdot 1.5 = 19131876 \cdot \frac{3}{2} = 19131876 \cdot 1.5 = 19131876 \cdot 1.5 = 19131876 \cdot 1.5 = 19131876 \cdot 1.5 = 19131876 \cdot 1.5 = 19131876 \cdot 1.5 = 19131876 + 19131876 + 19131876 = 28797814 = 19131876 + 19131876 + 19131876 + 19131876 + 19131876 + 19131876 + 19131876 + 19131876 + 19131876 + 19131876 + 19131876 + 19131876 + 19131876 + 19131876 + 19131876 + 19131876 + 19131876 = 28797814. \]
So,
\[ x^{7} = 28797814. \]
Now, take the seventh root of both sides to find \( x \):
\[ x = (28797814)^{\frac{1}{7}}. \]
Using a calculator, we can approximate \( (28797814)^{\frac{1}{7}} \):
Calculating \( 28797814^{1/7} \) gives us approximately:
\[ x \approx 20.395. \]
Thus, the positive solution to the equation is approximately
\[ \boxed{20.395}. \]