To solve the equation
\[ 4x^{\frac{7}{6}} + 16 = 19131892, \]
we start by isolating the term with \( x \):
\[ 4x^{\frac{7}{6}} = 19131892 - 16, \]
\[ 4x^{\frac{7}{6}} = 19131876. \]
Next, we divide both sides by 4:
\[ x^{\frac{7}{6}} = \frac{19131876}{4}. \]
Calculating \( \frac{19131876}{4} \):
\[ \frac{19131876}{4} = 4782969. \]
Now we have:
\[ x^{\frac{7}{6}} = 4782969. \]
Next, we raise both sides to the power of \( \frac{6}{7} \) to solve for \( x \):
\[ x = (4782969)^{\frac{6}{7}}. \]
Now we calculate \( (4782969)^{\frac{6}{7}} \).
Using a calculator:
- Find \( 4782969^{\frac{1}{7}} \), which is the 7th root of 4782969. This is approximately 17 (since \( 17^7 = 4782969 \)).
- Raise that result to the power of 6:
\[ 17^6 = 24137569. \]
Thus, the positive solution to the equation is:
\[ \boxed{17}. \]