What is the least integer whose square roof is an irrational number between 5 and 7
AAAaannndd the bot gets it wrong yet again!
√3 is not between 5 and 6.
Since 5^2 = 25 and 6^2 = 36,
26 is the smallest integer whose square root is between 5 and 7.
I apologize for my previous mistake. You are correct that the least integer whose square root is an irrational number between 5 and 7 is 26. Thank you for correcting me.
To find the least integer whose square root is an irrational number between 5 and 7, we need to determine the integer values that fall within this range and then test them to see if their square root is irrational.
Let's start by finding the square roots of integers between 5 and 7:
For 5: √5 ≈ 2.236
For 6: √6 ≈ 2.449
For 7: √7 ≈ 2.646
We can see that the square roots of all these numbers are irrational. However, none of them are integers.
To find an integer whose square root falls within the desired range, we need to keep increasing the number until we find one that satisfies the condition.
Let's continue checking the next few integers:
For 8: √8 ≈ 2.828
For 9: √9 = 3
The square root of 9 is an integer, but it is not within the desired range between 5 and 7.
Continuing further:
For 10: √10 ≈ 3.162
For 11: √11 ≈ 3.317
For 12: √12 ≈ 3.464
For 13: √13 ≈ 3.606
The square roots of 10, 11, 12, and 13 are all irrational, but none of them are integers.
Continuing this process, we find:
For 14: √14 ≈ 3.742
For 15: √15 ≈ 3.873
For 16: √16 = 4
Finally, we have found the least integer whose square root is an irrational number between 5 and 7: the number is 16.
To find the least integer whose square root is an irrational number between 5 and 7, we need to determine the smallest perfect square greater than 5 and the largest perfect square less than 7.
The perfect squares that are greater than 5 are:
4^2 = 16
Since the square root of 16 is 4, it is not an irrational number between 5 and 7.
The perfect squares that are less than 7 are:
2^2 = 4
3^2 = 9
Since the square root of 4 is 2, it is not an irrational number between 5 and 7. However, the square root of 9 is 3, which is an irrational number between 5 and 7.
Therefore, the least integer whose square root is an irrational number between 5 and 7 is 9.
We can begin by finding the square roots of 5 and 7 to get an idea of which integers to try. We get:
√5 ≈ 2.236
√7 ≈ 2.646
This means that the desired irrational number is between 2 and 3.
Now, we can start trying squares of integers between 2 and 3 to see which one gives an irrational number between 5 and 7:
2^2 = 4, which is too small
3^2 = 9, which is too big
4^2 = 16, which is definitely too big
5^2 = 25, which is outside the range we want
Therefore, the least integer whose square root is an irrational number between 5 and 7 is 3.