To order the numbers [tex]\(\sqrt{70}, -8.\overline{6}, \frac{25}{3}, -3\frac{2}{3}\)[/tex] from least to greatest, we must first approximate or exactly determine each value.
1. Approximate each value:
- [tex]\(\sqrt{70}\)[/tex]: To find this, we calculate [tex]\(\sqrt{70}\)[/tex]. This is approximately [tex]\(8.37\)[/tex].
- [tex]\(-8.\overline{6}\)[/tex]: This is a repeating decimal [tex]\(-8.6666\ldots\)[/tex].
- [tex]\(\frac{25}{3}\)[/tex]: This division results in approximately [tex]\(8.33\)[/tex].
- [tex]\(-3\frac{2}{3}\)[/tex]: This can be converted to a decimal by dividing the fraction:
[tex]\[
-3\frac{2}{3} = -3 - \frac{2}{3} = -3 - 0.6666\ldots = -3.6666\ldots
\][/tex]
2. Comparing the values:
- [tex]\(\sqrt{70} \approx 8.37\)[/tex]
- [tex]\( -8.\overline{6} = -8.6666\ldots \)[/tex]
- [tex]\(\frac{25}{3} \approx 8.33\)[/tex]
- [tex]\(-3\frac{2}{3} = -3.6666\ldots\)[/tex]
3. Order from least to greatest:
- First, compare the negative numbers: [tex]\(-8.6666\ldots\)[/tex] is less than [tex]\(-3.6666\ldots\)[/tex].
- Next, compare the positive numbers: [tex]\(8.33\)[/tex] (from [tex]\(\frac{25}{3}\)[/tex]) is less than [tex]\(8.37\)[/tex] (from [tex]\(\sqrt{70}\)[/tex]).
Putting it all together:
[tex]\[
-8.\overline{6}, -3 \frac{2}{3}, \frac{25}{3}, \sqrt{70}
\][/tex]
So, the numbers in order from least to greatest are:
[tex]\[
-8.\overline{6}, -3 \frac{2}{3}, \frac{25}{3}, \sqrt{70}
\][/tex]
