answer:1. **Rewrite the given equation**: Starting from the equation log_{3x}729 = x, we can rewrite it using the definition of logarithms: [ (3x)^x = 729 ] 2. **Analyze the equation**: We know that x must be positive for the base 3x to be valid and positive. 3. **Trial and error method**: Testing positive integer values of x: - For x = 3, calculate: [ (3 cdot 3)^3 = 9^3 = 729 ] - This matches the right-hand side of the equation, so x = 3 is a solution. 4. **Uniqueness of the solution**: The function (3x)^x increases as x increases (for x > 0), as both the base 3x and exponent x increase. Since a match at x = 3 gives 729, no larger value of x can satisfy the equation as it would result in a value greater than 729. 5. **Characterize the solution**: x = 3: - 3 is an integer. - 3 is not a perfect square (as the square root of 3 is not an integer). - 3 is not a perfect cube (as the cube root of 3 is not an integer). 6. **Conclusion**: Since x = 3 is an integer, not a perfect square, and not a perfect cube, the correct answer is: [ textbf{(A) text{A non-square, non-cube integer}} ] The final answer is boxed{textbf{(A)}}

answer:From |z+3+4i| leq 2, we know its geometric meaning is: The set of points in the complex plane whose distance to the point (-3, -4) is less than or equal to 2. The distance from (-3, -4) to the origin is: 5 Therefore, the maximum value of |z| is: 5 + 2 = 7 Hence, the answer is boxed{7}.

answer:1. **Convert the numeral representations to decimal:** The numeral 65 in base c can be expressed as 6c + 5 in decimal. Similarly, the numeral 56 in base d can be expressed as 5d + 6 in decimal. 2. **Equality for decimal values:** Since both represent the same number, we have: [ 6c + 5 = 5d + 6 ] 3. **Simplify the equation:** Rearranging gives: [ 6c - 5d = 1 ] or equivalently, [ 5d = 6c - 1 ] Solving for d, we get: [ d = frac{6c - 1}{5} ] 4. **Determine valid values for c and d:** For d to be an integer, 6c - 1 must be divisible by 5. We can express this condition as: [ 6c - 1 equiv 0 pmod{5} ] Simplifying, we find: [ 6c equiv 1 pmod{5} ] [ c equiv 1 pmod{5} ] The smallest positive integer satisfying this condition is c = 6. 5. **Check the validity of the bases:** Substituting c = 6 into the equation for d: [ d = frac{6 cdot 6 - 1}{5} = frac{36 - 1}{5} = frac{35}{5} = 7 ] 6. **Calculate c + d:** [ c + d = 6 + 7 = 13 ] Converting 13 to Roman numerals, we find mathrm{XIII}. The final answer is boxed{(B) XIII}

answer:To find the 80% percentile of the given data set, we first calculate the rank of the 80% percentile for 10 data points. This calculation is done as follows: - We multiply the total number of data points by 80%: 10 times 80% = 10 times 0.8 = 8. This means we are looking for the value that separates the lowest 80% of the data from the highest 20%. Since the calculation results in the number 8, which is not a whole number, we take the average of the 8^{th} and 9^{th} data points in our sorted set to find the 80% percentile. The sorted data set is 3, 4, 5, 5, 6, 7, 7, 8, 9, 10. Here, the 8^{th} data point is 8 and the 9^{th} data point is 9. The formula to calculate the 80% percentile in this case is: frac{1}{2} times (8^{th} text{ data point} + 9^{th} text{ data point}) = frac{1}{2} times (8 + 9) = frac{1}{2} times 17 = 8.5. Thus, the 80% percentile of this data set is 8.5. Therefore, the correct answer is boxed{C}.