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The concept of half-life is crucial in the realm of nuclear physics and particularly in understanding radioactive decay. It is the time required for half of the radioactive nuclei in a sample to undergo decay, transforming into a different element or isotope. This predictable rate of decay allows scientists to approximate the age of substances, the effectiveness of medical treatments, and even the time for a radioactive substance to become safe.

In practical terms, after one half-life, 50% of the original radioactive nuclei will have decayed. After two half-lives, 25% remain, and this pattern continues, halving with each consecutive half-life period. The exercise given illustrates this point by comparing the half-lives of two radioactive substances, A and B, each with different half-life times. Over these intervals, A and B will have different proportions of their original nuclei remaining, which leads us to use this concept to solve the problem of finding the ratio of decayed numbers of A and B nuclei.

The radioactive decay formula is a mathematical representation of the rate at which unstable atomic nuclei lose energy by emitting radiation. The standard formula used to determine the remaining number of radioactive nuclei after a certain amount of time is:
\[\begin{equation}N(t) = N_0 \times \bigg(\frac{1}{2}\bigg)^{\frac{t}{T}}\end{equation}\]
Here, \(N(t)\) is the number of nuclei remaining after time \(t\), \(N_0\) is the original number of nuclei, \(T\) is the half-life of the substance, and \(t\) is the time elapsed. When applying this formula to our problem, we find the ratio of decayed nuclei by first calculating how much of each substance remains and then determining what fraction has decayed. Understanding and correctly applying this formula is key to solving many problems related to the stability and longevity of radioactive materials.

To comprehend nuclei decay calculations, one must first grasp the radioactive decay formula and half-life concept. By integrating these concepts, we can calculate both the remaining and decayed nuclei after a certain period. In our textbook exercise, it's essential to calculate the decayed nuclei for two elements to find their decay ratio after 80 minutes.

The step-by-step solution walks through this process, demonstrating how to subtract the remaining nuclei from the initial number to get the decayed count. The final step involves simplifying a ratio, which is a common necessity in such problems, and recognizes the importance of algebraic manipulation in nuclear chemistry calculations. The practice of nuclear decay calculations equips students with the skill to predict and analyze radioactive elements' behavior over time, a fundamental aspect of both basic understanding and application in various scientific fields.