Question

Which of the following is an irrational number between 4 and 8?

Show your work.

(1 point)
Responses

17−−√
square root of 17

12−−√
square root of 12

5–√
square root of 5

8–√

give me a simple explanation

Answer 1

To find an irrational number between 4 and 8, we can start by calculating the square roots given in the options and see which one lies in that range.

1. **\(\sqrt{17}\)**:
- We know that \(4^2 = 16\) and \(5^2 = 25\).
- Since \(16 < 17 < 25\), we have \(4 < \sqrt{17} < 5\).
- Therefore, \(\sqrt{17}\) is **between 4 and 5**.

2. **\(\sqrt{12}\)**:
- We know that \(3^2 = 9\) and \(4^2 = 16\).
- Since \(9 < 12 < 16\), we have \(3 < \sqrt{12} < 4\).
- Therefore, \(\sqrt{12}\) is **between 3 and 4**.

3. **\(\sqrt{5}\)**:
- We know that \(2^2 = 4\) and \(3^2 = 9\).
- Since \(4 < 5 < 9\), we have \(2 < \sqrt{5} < 3\).
- Therefore, \(\sqrt{5}\) is **between 2 and 3**.

4. **\(\sqrt{8}\)**:
- We know that \(2^2 = 4\) and \(3^2 = 9\).
- Since \(4 < 8 < 9\), we have \(2 < \sqrt{8} < 3\).
- Therefore, \(\sqrt{8}\) is **between 2 and 3**.

None of option 2 or 3 contained irrational numbers in the desired range. The only one that is an irrational number and between 4 and 8 is \(\sqrt{17}\).

So, the answer is:

- **\(\sqrt{17}\)**, which is an irrational number between 4 and 8.