1. The problem involves evaluating and understanding several mathematical expressions: $e^6$, $\log_6(6)$, $\infty$, $\int_1^2 x \, dx$, $\sum_{i=0}^4 i$, $i$, $3!$, $\sqrt{21}$, and $y^x$. 2. Evaluate each expression step-by-step: - $e^6$ is the exponential function with base $e$ raised to the power 6. - $\log_6(6)$ is the logarithm base 6 of 6, which equals 1 because $6^1=6$. - $\infty$ represents infinity, a concept rather than a number. - $\int_1^2 x \, dx$ is the definite integral of $x$ from 1 to 2. - $\sum_{i=0}^4 i$ is the sum of integers from 0 to 4. - $i$ is typically the imaginary unit, $i=\sqrt{-1}$. - $3!$ is the factorial of 3, which is $3 \times 2 \times 1 = 6$. - $\sqrt{21}$ is the square root of 21. - $y^x$ is a general exponential expression with base $y$ and exponent $x$. 3. Calculate the integral: $$\int_1^2 x \, dx = \left[ \frac{x^2}{2} \right]_1^2 = \frac{2^2}{2} - \frac{1^2}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} = 1.5$$ 4. Calculate the sum: $$\sum_{i=0}^4 i = 0 + 1 + 2 + 3 + 4 = 10$$ 5. Summarize the results: - $e^6$ remains as is (approximately 403.4288). - $\log_6(6) = 1$. - $\infty$ is a concept, no numeric value. - $\int_1^2 x \, dx = 1.5$. - $\sum_{i=0}^4 i = 10$. - $i$ is the imaginary unit. - $3! = 6$. - $\sqrt{21} \approx 4.5826$. - $y^x$ is a general expression. This completes the evaluation and explanation of the given expressions.