Question

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

[AI-generated answer — Bot GPT 3.5]

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.

Answer 2

[AI-generated answer — Step-by-Step Bot]

To find a rational number with a denominator of 7 that is between √7 and √8, we can first evaluate the decimal approximations of √7 and √8:

√7 ≈ 2.645751311...
√8 ≈ 2.828427125...

Since we want a rational number with a denominator of 7, we need to find where this fraction will fit between these decimal approximations.

The first step is to find a common denominator. To do this, we will multiply the two denominators together:
1 * 7 = 7

Next, we can convert the two decimal approximations to fractions with the common denominator of 7:
√7 ≈ 2.645751311... ≈ 2.645751311... * 7/7 ≈ 18.52353718.../7
√8 ≈ 2.828427125... ≈ 2.828427125... * 7/7 ≈ 19.79998992.../7

Now, we can simplify these fractions to find the closest fraction with a denominator of 7 between them:
√7 ≈ 18.52353718.../7 ≈ 565/7
√8 ≈ 19.79998992.../7 ≈ 588/7

Since the decimal approximations are not exact, we can see that the fraction 565/7 is closer to √7, while the fraction 588/7 is closer to √8.

Therefore, the rational number with a denominator of 7 that is between √7 and √8 as an improper fraction is 565/7.

Answer 3

[AI-generated answer — Explain Bot]

To find a rational number with a denominator of 7 between √7 and √8, we need to consider the numbers that are multiples of √7 and √8.

Let's first simplify the square roots:
√7 ≈ 2.65
√8 ≈ 2.83

We want a number with a denominator of 7, so we need to find a multiple of 7 that lies between 2.65 and 2.83.

Let's consider the range of multiples of 7:
2 * 7 = 14
3 * 7 = 21
4 * 7 = 28
...
8 * 7 = 56

Looking at this range, let's find the multiple of 7 that is closest to √8, which is 2.83.

The closest multiple to 2.83 is 3, as 3 * 7 = 21.

Therefore, the rational number with a denominator of 7 between √7 and √8 is 21/7.

However, since 21/7 can be simplified, let's express it as an improper fraction:
21/7 = 3.

So, the rational number in the form of an improper fraction is 3.