Question

Which of the following is an irrational number between 5 and 7: 6–√
, 12−−√
, 27−−√
, 36−−√
?(1 point)
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Answer 1

To determine which of the given options is an irrational number between 5 and 7, we need to analyze each square root.

1. **\(6 - \sqrt{6}\)**:
- First, we find \(\sqrt{6}\). Since \(\sqrt{6} \approx 2.45\), we have:
\[
6 - \sqrt{6} \approx 6 - 2.45 \approx 3.55
\]
- This value is not between 5 and 7, and \(6 - \sqrt{6}\) is not irrational, since it's a result of a rational operation.

2. **\(\sqrt{12}\)**:
- We find \(\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\). Since \(\sqrt{3} \approx 1.732\), we have:
\[
\sqrt{12} = 2 \times 1.732 \approx 3.464
\]
- This value is also not between 5 and 7, and \(\sqrt{12}\) is irrational.

3. **\(\sqrt{27}\)**:
- We compute \(\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}\), and since \(\sqrt{3} \approx 1.732\):
\[
\sqrt{27} = 3 \times 1.732 \approx 5.196
\]
- This value is between 5 and 7, and \(\sqrt{27}\) is also irrational.

4. **\(\sqrt{36}\)**:
- Here, \(\sqrt{36} = 6\).
- This value is not irrational (it's rational) and is exactly 6, which is between 5 and 7.

Considering the above, the only irrational number between 5 and 7 is:

\[
\sqrt{27}
\]

Therefore, the answer is \(\sqrt{27}\).