Question

What is a rational number with a denominator of 7 that is betwejavascript:void(0);en 7–√ and 8–√ . Write your answer as an improper fraction.(1 point)

Answer 1

can you simplify

Answer 2

To find a rational number with a denominator of 7 that is between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \), we first need to calculate the approximate values of these expressions.

1. Calculate \( 7 - \sqrt{7} \):
- The approximate value of \( \sqrt{7} \) is about \( 2.64575 \).
- Thus, \( 7 - \sqrt{7} \approx 7 - 2.64575 \approx 4.35425 \).

2. Calculate \( 8 - \sqrt{8} \):
- The approximate value of \( \sqrt{8} \) is \( 2\sqrt{2} \approx 2.82843 \).
- Therefore, \( 8 - \sqrt{8} \approx 8 - 2.82843 \approx 5.17157 \).

Now, we are looking for a rational number with a denominator of 7 that lies between approximately \( 4.35425 \) and \( 5.17157 \).

Rational numbers with a denominator of 7 can be expressed in the form \( \frac{n}{7} \), where \( n \) is an integer.

To find suitable integers \( n \):

- Multiply the lower bound \( 4.35425 \) by 7:
\[
4.35425 \times 7 \approx 30.48075.
\]
This suggests that \( n \) must be greater than 30.

- Multiply the upper bound \( 5.17157 \) by 7:
\[
5.17157 \times 7 \approx 36.201.
\]
This suggests that \( n \) must be less than 36.

Thus, \( n \) can take on the values \( 31, 32, 33, 34, 35, \) or \( 36 \).

Choosing any of these values gives us the rational numbers:
- \( \frac{31}{7} \approx 4.42857 \)
- \( \frac{32}{7} \approx 4.57143 \)
- \( \frac{33}{7} \approx 4.71429 \)
- \( \frac{34}{7} \approx 4.85714 \)
- \( \frac{35}{7} \approx 5.00000 \)
- \( \frac{36}{7} \approx 5.14286 \)

All of these values fall between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \).

So, we can take any of these fractions as our final answer. A suitable answer could be \( \frac{34}{7} \).

Therefore, the answer is:
\[
\frac{34}{7}
\]

Answer 3

The fraction \(\frac{34}{7}\) is already in its simplest form because 34 and 7 have no common factors other than 1.

To clarify:
- The numerator, 34, can be factored into \(2 \times 17\).
- The denominator, 7, is a prime number.

Since there are no common factors, \(\frac{34}{7}\) cannot be simplified further.

If you are looking for a different rational number with a denominator of 7, the other options provided previously (like \(\frac{31}{7}\), \(\frac{32}{7}\), etc.) are also already in simplest form.

If you have any other questions or need further clarification, feel free to ask!