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

Answer 1

To determine if these numbers are irrational, we need to check if they are non-repeating and non-terminating decimals.

For 6–√, we have $\sqrt{6}$ which is approximately 2.449 since $\sqrt{4}=2$ and $\sqrt{9}=3$. Since $\sqrt{6}$ is not a simplified fraction, it is irrational.

For 12–√, we have $\sqrt{12}$ which is approximately 3.464 since $\sqrt{9}=3$ and $\sqrt{16}=4$. Again, since $\sqrt{12}$ is not a simplified fraction, it is irrational.

For 27–√, we have $\sqrt{27}$ which is approximately 5.196 since $\sqrt{25}=5$ and $\sqrt{36}=6$. Once again, because $\sqrt{27}$ is not a simplified fraction, it is irrational.

For 36–√, we have $\sqrt{36}$ which is exactly 6. Since this is a simplified fraction, it is not an irrational number.

Therefore, the irrational number between 5 and 7 is $\boxed{\sqrt{6}}$.