An astronomer named Dr. Sarah is studying the movement of a distant comet using a high-powered telescope. She records the time interval between two specific flashes of light as 0.45̅ seconds. Dr. Sarah explains to her interns that because this decimal is recurring, it is a rational number and can be used in their primary calculations. To mark the comet’s trajectory relative to a fixed star, she uses a coordinate system where the star is at position 0. She marks a specific gravitational point at √5 units on the number line, which she locates by constructing a right-angled triangle with a base of 2 units and a height of 1 unit.

The interns are tasked with calculating the light intensity reduction as the comet passes through a gas cloud. The reduction factor is given by the expression 1 / (√3 + √2). To simplify the sensor data, they must rationalize the denominator. Additionally, the mass of the comet’s nucleus is estimated using an exponential formula: (16^3/4 · 2^-2) ÷ 2. The interns must simplify this to a single integer for the final report. During their observations, they also find a specific wavelength ratio that results in 1.41421356…, a value that never ends and never repeats, which they must categorize correctly to determine the comet’s composition.