Question

What is a rational number with a denominator of 7 that is between 7–√
and 8–√
. Write your answer as an improper fraction.(1 point)

Answers

Answered by i love miners
what the answer
Answered by GPT-4o mini
To find a rational number with a denominator of 7 that is between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \), we first need to evaluate the two expressions numerically.

1. Calculate \( 7 - \sqrt{7} \):
- \( \sqrt{7} \approx 2.64575 \) (approximately).
- \( 7 - \sqrt{7} \approx 7 - 2.64575 \approx 4.35425 \).

2. Calculate \( 8 - \sqrt{8} \):
- \( \sqrt{8} = 2\sqrt{2} \approx 2.82843 \).
- \( 8 - \sqrt{8} \approx 8 - 2.82843 \approx 5.17157 \).

Now we note that we want a rational number with a denominator of 7 that lies between approximately \( 4.35425 \) and \( 5.17157 \).

To express this as a fraction with a denominator of 7, we can convert the decimal bounds into fractions:

- For \( 4.35425 \):
- \( 4.35425 \times 7 \approx 30.48075 \).
- The closest whole number is 30, so \( \frac{30}{7} \approx 4.28571 \), which is less than \( 4.35425 \).

- For \( 5.17157 \):
- \( 5.17157 \times 7 \approx 36.201 \).
- The closest whole number is 36, so \( \frac{36}{7} \approx 5.14286 \), which is greater than \( 5.17157 \).

A rational number with a denominator of 7 that falls between \( \frac{31}{7} \) and \( \frac{35}{7} \) is \( \frac{32}{7} \), \( \frac{33}{7} \), or \( \frac{34}{7} \).

Calculating:
- \( \frac{32}{7} \approx 4.57143 \) (between).
- \( \frac{33}{7} \approx 4.71429 \) (between).
- \( \frac{34}{7} \approx 4.85714 \) (between).

Thus, one possible rational number is:

\[
\frac{33}{7}
\]

This is between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \) as desired.

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