1. **State the problem:** We need to show that there is no constant term (term independent of $x$) in the expansion of $$\left(3x^8 - \frac{3}{x^{15}}\right)^{10}.$$\n\n2. **Recall the binomial expansion formula:** For any $a$ and $b$, and integer $n$, $$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k.$$\n\n3. **Apply to our expression:** Let $a = 3x^8$ and $b = -\frac{3}{x^{15}}$. Then the general term is\n$$T_{k+1} = \binom{10}{k} (3x^8)^{10-k} \left(-\frac{3}{x^{15}}\right)^k.$$\n\n4. **Simplify the general term:**\n$$T_{k+1} = \binom{10}{k} 3^{10-k} x^{8(10-k)} (-1)^k 3^k x^{-15k} = \binom{10}{k} (-1)^k 3^{10} x^{80 - 8k - 15k} = \binom{10}{k} (-1)^k 3^{10} x^{80 - 23k}.$$\n\n5. **Find the constant term condition:** The term is independent of $x$ if the exponent of $x$ is zero, so\n$$80 - 23k = 0.$$\n\n6. **Solve for $k$:**\n$$23k = 80 \implies k = \frac{80}{23}.$$\n\n7. **Interpretation:** Since $k$ must be an integer between $0$ and $10$, and $\frac{80}{23}$ is not an integer, there is no integer $k$ that makes the term independent of $x$.\n\n**Conclusion:** There is no constant term in the expansion because no term has $x^0$ power.