answer:Let's first multiply the equation z^3 - z + 1 = 0 by z^2 + z + 1: [ z^3 - z + 1 = 0 ] [ (z^3 - z + 1)(z^2 + z + 1) = 0 ] Expanding and simplifying: [ z^5 + z^4 + z^3 + z^3 - z^2 - z + z^2 + z + 1 = 0 ] [ z^5 + z^4 + 2z^3 - z + 1 = 0 ] Next, adding z^5 - 1 = 0 to the equation: [ (z^5 + z^4 + 2z^3 - z + 1) + (z^5 - 1) = 0 ] [ 2z^5 + z^4 + 2z^3 - z + 2 = 0 ] [ z^5 - 1 = (z-1)(z^4 + z^3 + z^2 + z + 1) = 0 ] Thus, z^5 = 1, implying that every root of z^3 - z + 1 = 0 is a 5^{text{th}} root of unity. Factoring z^5 - 1 as: [ (z-1)(z^4 + z^3 + z^2 + z + 1) = 0 ] The 5^{text{th}} roots of unity are e^{0}, e^{2pi i/5}, e^{4pi i/5}, e^{6pi i/5}, e^{8pi i/5}. So, the smallest positive integer n such that (e^{2pi i/5})^n = 1 is 5. Therefore, the smallest possible value of n is boxed{5}.

answer:Given the properties of trapezoid PQRS: - [PQT] = 75 square units - [PST] = 45 square units Since PQRS is a trapezoid with PQ parallel RS, triangles PTR and QTS share a base PT and have the same altitude from PT to RS, implying [PTR] = [QTS]. To find the area of triangle QTS, note that triangles PQT and PST share an altitude from P to line QS, thus their areas are proportional to the lengths of their bases QT and TS, respectively. Therefore, QT/TS = [PQT]/[PST] = 75/45 = 5/3. Since the ratio of the areas of similar triangles is the square of the ratio of their sides, [QTS]/[PQT] = (TS/QT)^2 = (3/5)^2 = 9/25. Therefore, [QTS] = (9/25)[PQT] = (9/25)(75) = 27 square units. The area of trapezoid PQRS is then [PQT] + [PST] + [PTR] + [QTS] = 75 + 45 + 45 + 27 = boxed{192} square units.

answer:Since 40^circ < 50^circ, it follows that sin 40^circ < sin 50^circ. Therefore, the answer is <. This can be concluded from the fact that when 0 < alpha < 90^circ, sin alpha increases as alpha increases, which leads to sin 40^circ < sin 50^circ. This question examines the increasing nature of acute angle trigonometric functions: for the sine function, when 0 < alpha < 90^circ, sin alpha increases as alpha increases. Thus, the final answer is boxed{<}.

answer:**Analysis:** This question mainly examines the knowledge related to Digital Earth. It tests the application of geographic information technology. Digital Earth allows us to navigate the internet, but it does not enable physical travel, hence option ① is incorrect. Therefore, the answer is boxed{B}. **Review:** This question primarily tests knowledge about Digital Earth. When answering this question, it is important to closely integrate the knowledge learned and analyze carefully.