Question

to two decimal places, \\(\sqrt{82}\\) must lie between 9.05 and 9.06 because \\(9.05^2 = 81.9025\\) and \\(9.06^2 = 82.0836\\), and 82 lies between these values. drag \\(\sqrt{82}\\) based on your estimate above: number line from 9 to 9.1 with \\(\sqrt{82}\\) marked between 9.05 and 9.06 answer attempt 1 out of 2 it has been demonstrated that \\(\sqrt{82}\\) is between 9.05 and 9.06. if we were to continue this process, it would dropdown because \\(\sqrt{82}\\) is dropdown. submit answer

Explanation:

To determine the behavior of approximating \(\sqrt{82}\), we know that \(\sqrt{82}\) is an irrational number (since 82 is not a perfect square, and the square root of a non - perfect square positive integer is irrational). When we use the method of successive approximation (like the one used to show it is between 9.05 and 9.06, by squaring numbers and comparing to 82), we are using a process similar to the bisection method or just successive refinement of the interval. Since \(\sqrt{82}\) is irrational, its decimal expansion is non - repeating and non - terminating. So, if we continue the process of approximating it (finding more and more decimal places), we will be able to get closer and closer to the actual value of \(\sqrt{82}\) (this is the nature of approximating irrational numbers; we can keep refining our estimate to be as close as we want to the true value).

For the first dropdown (the action of the process): The process of approximating \(\sqrt{82}\) (by looking at smaller and smaller intervals around it) will "converge to a unique value" (or a similar phrase about getting closer to the actual value) because we are narrowing down the interval around the irrational number. For the second dropdown, \(\sqrt{82}\) is "irrational" (since 82 is not a perfect square, \(\sqrt{82}\) cannot be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q
eq0\), so it is irrational).

Answer:

First dropdown: converge to a unique value (or similar approximation - related convergence phrase)
Second dropdown: irrational