Find the average value of the function $k(x)=7 x^{3}+9 x$ on the interval $[4,8]$.

Step 1 ：The average value of a function \(f(x)\) on the interval \([a, b]\) is given by the formula: \[\frac{1}{b-a} \int_{a}^{b} f(x) dx\]

Step 2 ：We are asked to find the average value of the function \(k(x)=7x^3+9x\) on the interval \([4,8]\).

Step 3 ：To do this, we need to integrate the function from 4 to 8, and then divide by the length of the interval, which is \(8-4=4\).

Step 4 ：The integral of \(k(x)=7x^3+9x\) from 4 to 8 is 27744.

Step 5 ：Dividing this by the length of the interval, 4, we get 6936.

Step 6 ：Final Answer: The average value of the function \(k(x)=7 x^{3}+9 x\) on the interval \([4,8]\) is \(\boxed{1734}\).