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Show an example of an irrational number between 9 and 11.(1 point)

Responses

100−−−√
start root 100 end root

10−−√
start root 10 end root

20−−√
start root 20 end root

99−−√

What is the least integer whose square root is an irrational number between 5 and 7?(1 point)
Responses

26
26

25
25

6
6

13

Show an example of a rational number that is between 10−−√
and 11−−√
.(1 point)
Responses

3.1
3.1

165
Start Fraction 16 over 5 end fraction

3.4
3.4

10/3

Which irrational number is between 14−−√
and 18−−√
?(1 point)
Responses

15−−√
start root 15 end root

19−−√
start root 19 end root

10−−√
start root 10 end root

13−−√

How many irrational numbers are there between 11−−√
and 17−−√
?(1 point)
Responses

an infinite number
an infinite number

six numbers
six numbers

twenty-eight numbers
twenty-eight numbers

four numbers

1 answer

Let's go through each question one by one.

  1. Show an example of an irrational number between 9 and 11.

    • The possible options include:
      • \( \sqrt{100} = 10 \) (rational)
      • \( \sqrt{10} \approx 3.16 \) (not between 9 and 11)
      • \( \sqrt{20} \approx 4.47 \) (not between 9 and 11)
      • \( \sqrt{99} \approx 9.95 \) (irrational)
    • Answer: \( \sqrt{99} \)

  2. What is the least integer whose square root is an irrational number between 5 and 7?

    • The square roots we are considering:
      • \( 5^2 = 25 \)
      • \( 7^2 = 49 \)
    • The least integer greater than 25 whose square root is irrational and less than 49 is 26.
    • Answer: 26

  3. Show an example of a rational number that is between \( \sqrt{10} \) and \( \sqrt{11} \).

    • We know:
      • \( \sqrt{10} \approx 3.16 \)
      • \( \sqrt{11} \approx 3.32 \)
    • A rational number between these two is \( \frac{16}{5} = 3.2 \).
    • Answer: \( \frac{16}{5} \)

  4. Which irrational number is between \( \sqrt{14} \) and \( \sqrt{18} \)?

    • We know:
      • \( \sqrt{14} \approx 3.74 \)
      • \( \sqrt{18} \approx 4.24 \)
    • An irrational number between these two is \( \sqrt{15} \).
    • Answer: \( \sqrt{15} \)

  5. How many irrational numbers are there between \( \sqrt{11} \) and \( \sqrt{17} \)?

    • Since between any two real numbers (including square roots), there are infinitely many numbers, including both rational and irrational.
    • Answer: an infinite number

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