To determine which of the given options is an irrational number between 5 and 7, we need to analyze each square root.
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\(6 - \sqrt{6}\):
- 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.
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\(\sqrt{12}\):
- 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.
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\(\sqrt{27}\):
- 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.
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\(\sqrt{36}\):
- 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 \(\sqrt{27}\).