Question
What is the least integer whose square root is an irrational number between 5 and
7? (1 point)
• 13
• 25
• 26
7? (1 point)
• 13
• 25
• 26
Answers
GPT-4o mini
To find the least integer whose square root is an irrational number between 5 and 7, we start by determining the range for the square of the irrational number.
1. Calculate the squares of 5 and 7:
- \( 5^2 = 25 \)
- \( 7^2 = 49 \)
2. We seek an integer \( n \) such that \( 25 < n < 49 \) and \( \sqrt{n} \) is irrational.
3. The integers between 25 and 49 are: 26, 27, 28, ..., 48.
4. An integer \( n \) will have an irrational square root if it is not a perfect square. The perfect squares between 25 and 49 are:
- \( 25 = 5^2 \)
- \( 36 = 6^2 \)
- \( 49 = 7^2 \)
5. This means the perfect squares we need to avoid are 25 and 36. The numbers 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, and 48 are candidates.
6. The smallest integer in that range is 26.
Thus, the least integer whose square root is an irrational number between 5 and 7 is \( \boxed{26} \).
1. Calculate the squares of 5 and 7:
- \( 5^2 = 25 \)
- \( 7^2 = 49 \)
2. We seek an integer \( n \) such that \( 25 < n < 49 \) and \( \sqrt{n} \) is irrational.
3. The integers between 25 and 49 are: 26, 27, 28, ..., 48.
4. An integer \( n \) will have an irrational square root if it is not a perfect square. The perfect squares between 25 and 49 are:
- \( 25 = 5^2 \)
- \( 36 = 6^2 \)
- \( 49 = 7^2 \)
5. This means the perfect squares we need to avoid are 25 and 36. The numbers 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, and 48 are candidates.
6. The smallest integer in that range is 26.
Thus, the least integer whose square root is an irrational number between 5 and 7 is \( \boxed{26} \).