Growth Rate 7Edae5

1. **State the problem:** We need to determine which function grows faster as $n$ becomes very large: $n^{10!}$ or $0.0000001^n$. 2. **Recall the functions:** - $n^{10!}$ is a polynomial function with a very large exponent $10! = 3628800$. - $0.0000001^n$ is an exponential function with base $0.0000001$ (which is less than 1). 3. **Important rule:** For large $n$, exponential functions with base less than 1 approach zero, i.e., $0 < b < 1 \implies b^n \to 0$ as $n \to \infty$. 4. **Behavior of $n^{10!}$:** As $n$ grows, $n^{10!}$ grows very large (polynomial growth). 5. **Behavior of $0.0000001^n$:** Since $0.0000001 < 1$, $0.0000001^n$ becomes very small and approaches zero as $n$ increases. 6. **Conclusion:** For sufficiently large $n$, $n^{10!}$ grows without bound, while $0.0000001^n$ shrinks to zero. **Final answer:** $n^{10!}$ grows at a faster rate than $0.0000001^n$. Therefore, the correct choice is **$n^{10!}$**.