To estimate [tex]\(\sqrt{10}\)[/tex] to the nearest hundredth, follow these steps:
1. First, recognize the perfect squares around 10. We know that [tex]\(3^2 = 9\)[/tex] and [tex]\(4^2 = 16\)[/tex]. Therefore, [tex]\(\sqrt{10}\)[/tex] is between 3 and 4.
2. For more precision, we can narrow it down further by trying numbers between 3 and 4.
For instance, let's compare 3.1 and 3.2:
- [tex]\(3.1^2 = 9.61\)[/tex]
- [tex]\(3.2^2 = 10.24\)[/tex]
Since 9.61 is less than 10 and 10.24 is greater than 10, [tex]\(\sqrt{10}\)[/tex] is between 3.1 and 3.2.
3. Continuing this method, we further narrow it down by comparing values closer to 3.16. We can check the values 3.15 and 3.17:
- [tex]\(3.15^2 = 9.9225\)[/tex]
- [tex]\(3.16^2 = 9.9856\)[/tex]
- [tex]\(3.17^2 = 10.0489\)[/tex]
Since [tex]\(9.9856\)[/tex] is very close to 10 and [tex]\(10.0489\)[/tex] is just higher than 10, it’s clear that [tex]\(\sqrt{10}\)[/tex] is very close to 3.16.
4. Given this detail, we can confidently estimate that [tex]\(\sqrt{10}\)[/tex] to the nearest hundredth is approximately [tex]\(3.16\)[/tex].
