1. Let's analyze the expression $\infty^0$ where $0$ is an absolute value, not a limit. 2. By definition, any nonzero number raised to the power of 0 is 1. However, $\infty$ is not a real number but a concept representing unbounded growth. 3. In strict terms, $\infty^0$ is considered an indeterminate form because it depends on the limiting process that leads to these values. 4. But if we treat $\infty$ as an absolute value (not a limit), the expression is not well-defined in standard arithmetic, so it is indeterminate. 5. Now, consider $1^\infty$ where 1 is an absolute value, not a limit. 6. Since 1 raised to any finite power is always 1, and $\infty$ here is an absolute value, $1^\infty = 1$. 7. However, in limits, $1^\infty$ is an indeterminate form because the base approaches 1 and the exponent approaches infinity, which can lead to different results. 8. Summary: - $\infty^0$ with absolute values is indeterminate. - $1^\infty$ with absolute values equals 1.