what is a rational number with a denominator of 7 that is between \sqrt(7) and \sqrt(8). write your answer as an improper fraction

Answer 1

To find a rational number with a denominator of 7 between √7 and √8, we can start by simplifying √7 and √8:
√7 ≈ 2.646
√8 ≈ 2.828

Now, we need to find a fraction with a denominator of 7 that lies between 2.646 and 2.828. To achieve this, we can look for the nearest integers whose multiples of 7 fall between these two values.

The nearest multiples of 7 for 2.646 and 2.828 are:
2.646 ≈ 2.646 * 7 = 18.522 ≈ 18
2.828 ≈ 2.828 * 7 = 19.796 ≈ 20

So, we know that the fraction will be between 18/7 and 20/7. However, we need to express it as an improper fraction, meaning the numerator should be greater than the denominator.

To get an improper fraction, we multiply 19/7 (between 18/7 and 20/7) by 7/7:
19/7 * 7/7 = 133/49

Therefore, the rational number with a denominator of 7 that lies between √7 and √8 is 133/49.