Question

arrange the following real numbers in order from greatest to least. 0.9, 0.overline{9}, 0.9overline{7}, 0.97overline{7}, 0.99overline{7}, 0.79overline{7}, sqrt{0.97} drag each of the real numbers above to its correct location below. 0.99overline{7}, sqrt{0.97}, 0.overline{9}, 0.9overline{7}, 0.79overline{7}, 0.9, 0.97overline{7}

Explanation:

Step1: Analyze each number

• \(0.9 = 0.9\)
• \(0.\overline{9}=0.999\cdots\) (since the 9 repeats infinitely)
• \(0.\overline{97}=0.979797\cdots\) (97 repeats)
• \(0.9\overline{7}=0.9777\cdots\) (7 repeats after 9)
• \(0.99\overline{7}=0.99777\cdots\) (7 repeats after 99)
• \(0.7\overline{97}=0.79797\cdots\) (97 repeats after 7)
• \(\sqrt{0.97}\approx0.9849\) (calculate the square root of 0.97)

Step2: Compare the numbers

• \(0.\overline{9}=0.999\cdots\) is the largest.
• Next, \(0.99\overline{7}=0.99777\cdots\) (since \(0.999\cdots>0.99777\cdots\))
• Then, \(\sqrt{0.97}\approx0.9849\) (since \(0.99777\cdots>0.9849\))
• Next, \(0.\overline{97}=0.979797\cdots\) (since \(0.9849>0.979797\cdots\))
• Then, \(0.9\overline{7}=0.9777\cdots\) (since \(0.979797\cdots>0.9777\cdots\))
• Next, \(0.9\) (since \(0.9777\cdots>0.9\))
• Then, \(0.7\overline{97}=0.79797\cdots\) (since \(0.9>0.79797\cdots\))

Step3: Order from greatest to least

1. \(0.\overline{9}\)
2. \(0.99\overline{7}\)
3. \(\sqrt{0.97}\)
4. \(0.\overline{97}\)
5. \(0.9\overline{7}\)
6. \(0.9\)
7. \(0.7\overline{97}\)

Answer:

From greatest to least: \(0.\overline{9}\), \(0.99\overline{7}\), \(\sqrt{0.97}\), \(0.\overline{97}\), \(0.9\overline{7}\), \(0.9\), \(0.7\overline{97}\)