Which of the following is an irrational number between 5 and 7: 6–√, 12−√, 27−√, 36−√?
Pretty slow learner for an AI.
Many users type 6-√ when they mean √6
So, the question now reads:
Which of the following is an irrational number between 5 and 7: √6, √12, √27, √36?
Clearly, the answer is now √27, since 5^2 < 27 < 6^2
Apologies for misunderstanding the question earlier.
You are correct. Between 5 and 7, $\sqrt{27}$ is an irrational number since $5^2 = 25 < 27 < 36 = 6^2$. Thus, the answer is $\boxed{\sqrt{27}}$.
To determine which of the given numbers is an irrational number between 5 and 7, we need to check if the square root (√) of each number is irrational.
Let's calculate the square roots of the given numbers:
1. For 6–√:
√(6–√) ≈ 2.316
2. For 12−√:
√(12−√) ≈ 2.884
3. For 27−√:
√(27−√) ≈ 2.694
4. For 36−√:
√(36−√) ≈ 2.531
From the calculations, we can see that the square roots of all the given numbers are approximately rational numbers. Therefore, none of the given numbers (6–√, 12−√, 27−√, or 36−√) are irrational numbers between 5 and 7.
Since we know that $2^2 = 4 < 5$ and that $3^2=9>7,$ we can conclude that $2 < \sqrt{5} < 3,$ and $2 < \sqrt{7} < 3.$ Thus, $6 - \sqrt{5}$ and $6 - \sqrt{7}$ are both between $5$ and $7.$
Similarly, $12 - \sqrt{5}$ and $12 - \sqrt{7}$ are between $11$ and $13.$ Also, $27 - \sqrt{7}$ and $36 - \sqrt{7}$ are between $26$ and $28.$
Therefore, $\boxed{6 - \sqrt{5}}$ is an irrational number between $5$ and $7.$