Use the binomial theorem to find the term containing $x^{-33}$ in the expansion $\left(-3 x+\frac{1}{x^{9}}\right)^{7}$

Step 1 ：Given the binomial expansion \(\left(-3 x+\frac{1}{x^{9}}\right)^{7}\), we want to find the term containing \(x^{-33}\).

Step 2 ：We use the binomial theorem, which states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\). In this case, \(a = -3x\), \(b = \frac{1}{x^{9}}\), and \(n = 7\).

Step 3 ：We want to find the term where the power of \(x\) is \(-33\). This will occur when the powers of \(a\) and \(b\) in the binomial theorem sum to \(-33\).

Step 4 ：Let's denote the power of \(a\) as \(n-k\) and the power of \(b\) as \(k\). Since \(a = -3x\) and \(b = \frac{1}{x^{9}}\), the power of \(x\) in the term is \(n-k - 9k = -33\).

Step 5 ：Solving this equation, we find that the value of \(k\) that makes the power of \(x\) equal to \(-33\) in the term is \(4\).

Step 6 ：Now, we can substitute \(k = 4\) into the binomial theorem to find the coefficient of the term.

Step 7 ：The coefficient of the term is \(-945\).

Step 8 ：Final Answer: The term containing \(x^{-33}\) in the expansion \(\left(-3 x+\frac{1}{x^{9}}\right)^{7}\) is \(-945 x^{-33}\). Therefore, the coefficient of this term is \(\boxed{-945}\).