hich of the following is an irrational number between 5 and 7: 6–√ , 12−−√ , 27−−√ , 36−−√ ?

Answer 1

To determine which of the given options is an irrational number between 5 and 7, let's evaluate each option:

1. \( 6 - \sqrt{6} \):

   • First, we calculate \( \sqrt{6} \). The approximate value is about 2.45.
   • Thus, \( 6 - \sqrt{6} \approx 6 - 2.45 \approx 3.55 \) (which is not between 5 and 7).

2. \( \sqrt{12} \):

   • Simplifying, \( \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \).
   • The approximate value of \( \sqrt{3} \) is about 1.73.
   • Therefore, \( \sqrt{12} \approx 2 \times 1.73 \approx 3.46 \) (which is not between 5 and 7).

3. \( \sqrt{27} \):

   • Simplifying, \( \sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3} \).
   • Again, using the approximate value of \( \sqrt{3} \), we have \( 3\sqrt{3} \approx 3 \times 1.73 \approx 5.19 \) (which is between 5 and 7).

4. \( \sqrt{36} \):

   • Simplifying, \( \sqrt{36} = 6 \).
   • 6 is a rational number, so this does not satisfy the criteria.

From the analysis, \( \sqrt{27} \) or \( 3\sqrt{3} \) is the only option that is an irrational number and is between 5 and 7.

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