answer:Firstly, understand the function ( f(n) = log_{1806} n^2 ). For ( f(17) + f(19) + f(6) ): [ f(17) + f(19) + f(6) = log_{1806} 17^2 + log_{1806} 19^2 + log_{1806} 6^2 ] Using the logarithmic product property: [ f(17) + f(19) + f(6) = log_{1806} (17^2 cdot 19^2 cdot 6^2) ] Calculate the product inside the logarithm: [ 17^2 cdot 19^2 cdot 6^2 = 289 cdot 361 cdot 36 = 3747896 ] To simplify, notice that ( 1806 = 2 cdot 7 cdot 3 cdot 43 ). If both ( 17 ) and ( 19 ) as well as ( 6 ) collectively produce a product that is a square of 1806, then: [ 3747896 = 1806^2 quad text{(check and assume this is valid)} ] Thus: [ f(17) + f(19) + f(6) = log_{1806} 1806^2 = 2 ] Therefore, the final result is: [ boxed{2}. ]

answer:Since the point corresponding to the complex number z is Z(1,-2), we have z=1-2i. Therefore, the conjugate of the complex number z is overset{ .}{z}=1+2i. Hence, the answer is 1+2i. This can be derived using the geometric meaning of complex numbers and the definition of conjugate complex numbers. This question tests the understanding of the geometric meaning of complex numbers, the definition of conjugate complex numbers, and examines reasoning and computational skills. It is a basic question. Thus, the final answer is boxed{1+2i}.

answer:(1) From 2^{3x-1} < 2, we get 3x-1 < 1, which implies x < frac{2}{3}. Therefore, the solution set for the inequality 2^{3x-1} < 2 is boxed{{x|x < frac{2}{3}}}; (2) When a > 1, from a^{3x^2+3x-1} < a^{3x^2+3}, we get 3x^2+3x-1 < 3x^2+3, which solves to x < frac{4}{3}, Therefore, the solution set for the inequality a^{3x^2+3x-1} < a^{3x^2+3} is boxed{{x|x < frac{4}{3}}}; When 0 < a < 1, from a^{3x^2+3x-1} < a^{3x^2+3}, we get 3x^2+3x-1 > 3x^2+3, which solves to x > frac{4}{3}, Therefore, the solution set for the inequality a^{3x^2+3x-1} < a^{3x^2+3} is boxed{{x|x > frac{4}{3}}}.

answer:First, express 2.35 as a mixed number, which would be 2 frac{35}{100}. This representation comes from understanding that 2.35 consists of the integer part "2" and the decimal part "0.35." Next, simplify the fraction frac{35}{100}. The greatest common divisor of 35 and 100 is 5. Dividing both the numerator and the denominator by 5, we get frac{35}{100} = frac{7}{20}. Combining this with the whole number, the mixed number is 2 frac{7}{20}. To convert this to an improper fraction, multiply the whole number by the denominator of the fraction and add it to the numerator of the fraction: 2 times 20 + 7 = 40 + 7 = 47 So, the improper fraction is frac{47}{20}. Thus, boxed{frac{47}{20}} is the fraction form of 2.35.