To determine the most precise approximation of [tex]\(\sqrt{12}\)[/tex] among the given options, let's analyze each value:
1. Firstly, note that [tex]\(\sqrt{12}\)[/tex] is an irrational number, which means it has infinite non-repeating decimal places. Therefore, we will not get an exact value but an approximation.
2. Compare each option numerically to approximate [tex]\(\sqrt{12}\)[/tex]:
- Option A: [tex]\(3.0\)[/tex]
- This is a simple approximation and can be validated quickly. [tex]\(3.0^2 = 9\)[/tex], which is not very close to 12.
- Option B: [tex]\(3.446\)[/tex]
- Calculate [tex]\(3.446^2\)[/tex]:
[tex]\[
(3.446)^2 \approx 11.868
\][/tex]
- This is closer to 12 but not quite there.
- Option C: [tex]\(3.46\)[/tex]
- Calculate [tex]\(3.46^2\)[/tex]:
[tex]\[
(3.46)^2 \approx 11.9716
\][/tex]
- This is a closer approximation but still not perfect.
- Option D: [tex]\(3.464\)[/tex]
- Calculate [tex]\(3.464^2\)[/tex]:
[tex]\[
(3.464)^2 \approx 11.999296
\][/tex]
- This is very close to 12.
3. After comparing the closeness of each square to 12, the value [tex]\(3.464\)[/tex] provides the closest squared result to 12.
Therefore, the most precise approximation of [tex]\(\sqrt{12}\)[/tex] from the given options is:
[tex]\[
\boxed{3.464}
\][/tex]
