Given the function $C(n)=-14(n-6)^{2}(n+3)^{8}(n-7)$, its roots are and their multiplicities are

Step 1 ：Given the function \(C(n)=-14(n-6)^{2}(n+3)^{8}(n-7)\), we need to find its roots and their multiplicities.

Step 2 ：The roots of a polynomial are the values of \(n\) that make the polynomial equal to zero. The multiplicity of a root is the number of times it appears as a root, which is equivalent to the power of the factor in the polynomial.

Step 3 ：In this case, the roots are the values of \(n\) that make each factor equal to zero. The roots are \(n=6\), \(n=-3\), and \(n=7\).

Step 4 ：The multiplicities are the powers of the corresponding factors, which are \(2\), \(8\), and \(1\) respectively.

Step 5 ：Final Answer: The roots of the function \(C(n)=-14(n-6)^{2}(n+3)^{8}(n-7)\) are \(n=6\), \(n=-3\), and \(n=7\). Their multiplicities are \(2\), \(8\), and \(1\) respectively. So, the roots and their multiplicities are \(\boxed{(6,2), (-3,8), (7,1)}\).