To locate [tex]\(\sqrt{6.5}\)[/tex] on the number line, follow these comprehensive steps to understand its position:
1. Identify the interval:
- Recognize that [tex]\(6.5\)[/tex] is a bit more than [tex]\(6\)[/tex], and a bit less than [tex]\(7\)[/tex].
- The square roots of [tex]\(6\)[/tex] and [tex]\(7\)[/tex] thus form a range. We calculate square roots of these bounds to know our approximate range:
[tex]\[
\sqrt{6} \approx 2.449 \quad \text{and} \quad \sqrt{7} \approx 2.646
\][/tex]
2. Range verification:
- [tex]\(\sqrt{6.5}\)[/tex] will lie somewhere between [tex]\(\sqrt{6}\)[/tex] and [tex]\(\sqrt{7}\)[/tex]:
[tex]\[
2.449 < \sqrt{6.5} < 2.646
\][/tex]
3. Exact calculation details (result provided):
- The precise value of [tex]\(\sqrt{6.5}\)[/tex] is approximately [tex]\(2.5495097567963922\)[/tex].
4. Locate on the number line:
- First, mark the values [tex]\(2\)[/tex] and [tex]\(3\)[/tex] on the number line.
- Then, divide the interval between [tex]\(2\)[/tex] and [tex]\(3\)[/tex] into ten equal segments to mark the tenths positions such as [tex]\(2.1, 2.2, \ldots, 2.9\)[/tex].
5. Refine between 2.5 and 2.6:
- Narrow down between the relevant markings [tex]\(2.5\)[/tex] and [tex]\(2.6\)[/tex].
- [tex]\(2.5495097567963922\)[/tex] is slightly less than halfway between these two points.
6. Plot the point:
- Place a mark slightly less than halfway between [tex]\(2.5\)[/tex] and [tex]\(2.6\)[/tex] on the number line. This is where [tex]\(\sqrt{6.5}\)[/tex] is located, very close to [tex]\(2.55\)[/tex].
By following these steps, you can accurately estimate and plot [tex]\(\sqrt{6.5}\)[/tex] on a number line. Keep in mind, the exact position is close to [tex]\(2.55\)[/tex], giving us a precise location for [tex]\(\sqrt{6.5}\)[/tex].
