Numbers Between Rational Numbers Quick Check
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Question
What is the least integer whose square root is an irrational number between 5 and 7?(1 point)
The square root of 25 is 5, and the square root of 49 is 7. Since we are looking for the least integer whose square root is an irrational number between 5 and 7, we need to find the next integer after 49 whose square root is between 5 and 7. The next integer after 49 is 50, and its square root is approximately 7.07, which is between 5 and 7. Therefore, the least integer whose square root is an irrational number between 5 and 7 is 50.
To find the least integer whose square root is an irrational number between 5 and 7, we need to determine the perfect squares that are closest to 5 and 7.
First, check the square of the whole number closest to 5:
2^2 = 4
3^2 = 9
So, the square root of a perfect square between 4 and 9 will give us an irrational number between 2 and 3.
Next, check the square of the whole number closest to 7:
2^2 = 4
3^2 = 9
4^2 = 16
So, the square root of a perfect square between 9 and 16 will give us an irrational number between 3 and 4.
Since we want an irrational number between 5 and 7, we need to find the first perfect square larger than 9 whose square root falls between 5 and 7.
5^2 = 25
6^2 = 36
The square root of 25 is 5, but the square root of 36 is 6. Thus, the least integer whose square root is an irrational number between 5 and 7 is 6.