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) $$ Skip to navigation
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Answer
can u say it with out all the /{} []
Answered by
GPT-4o mini
To find a rational number with a denominator of 7 that lies between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \), we first need to evaluate the decimal values of these expressions.
1. **Calculate \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \)**:
- \( \sqrt{7} \approx 2.6457513110645906 \)
So, \( 7 - \sqrt{7} \approx 7 - 2.6457513110645906 \approx 4.3542486889354094 \).
- \( \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \approx 2.8284271247461903 \)
Thus, \( 8 - \sqrt{8} \approx 8 - 2.8284271247461903 \approx 5.1715728752538097 \).
2. **Now we need a rational number in the interval \( (4.35425, 5.17157) \) with a denominator of 7.**
3. **Forming a rational number with denominator 7**:
Let’s consider a rational number of the form \( \frac{n}{7} \) where \( n \) is an integer. We need to find \( n \) such that:
\[
4.35425 < \frac{n}{7} < 5.17157
\]
Multiplying all parts of the inequality by 7 (since 7 is positive), we get:
\[
7 \times 4.35425 < n < 7 \times 5.17157
\]
\[
30.46575 < n < 36.201
\]
The integers \( n \) that satisfy this inequality are \( 31, 32, 33, 34, 35, 36 \).
4. **Identifying suitable fractions**:
We can express these integers as fractions:
- \( \frac{31}{7} \approx 4.42857 \)
- \( \frac{32}{7} \approx 4.57143 \)
- \( \frac{33}{7} \approx 4.71429 \)
- \( \frac{34}{7} \approx 4.85714 \)
- \( \frac{35}{7} \approx 5.00000 \)
- \( \frac{36}{7} \approx 5.14286 \)
5. **Selecting a suitable fraction**:
All fractions from \( \frac{31}{7} \) to \( \frac{35}{7} \) are between \( 4.35425 \) and \( 5.17157 \).
We can choose any of these fractions. To provide one example:
\[
\frac{34}{7}
\]
Thus, the answer is:
\[
\boxed{\frac{34}{7}}
\]
1. **Calculate \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \)**:
- \( \sqrt{7} \approx 2.6457513110645906 \)
So, \( 7 - \sqrt{7} \approx 7 - 2.6457513110645906 \approx 4.3542486889354094 \).
- \( \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \approx 2.8284271247461903 \)
Thus, \( 8 - \sqrt{8} \approx 8 - 2.8284271247461903 \approx 5.1715728752538097 \).
2. **Now we need a rational number in the interval \( (4.35425, 5.17157) \) with a denominator of 7.**
3. **Forming a rational number with denominator 7**:
Let’s consider a rational number of the form \( \frac{n}{7} \) where \( n \) is an integer. We need to find \( n \) such that:
\[
4.35425 < \frac{n}{7} < 5.17157
\]
Multiplying all parts of the inequality by 7 (since 7 is positive), we get:
\[
7 \times 4.35425 < n < 7 \times 5.17157
\]
\[
30.46575 < n < 36.201
\]
The integers \( n \) that satisfy this inequality are \( 31, 32, 33, 34, 35, 36 \).
4. **Identifying suitable fractions**:
We can express these integers as fractions:
- \( \frac{31}{7} \approx 4.42857 \)
- \( \frac{32}{7} \approx 4.57143 \)
- \( \frac{33}{7} \approx 4.71429 \)
- \( \frac{34}{7} \approx 4.85714 \)
- \( \frac{35}{7} \approx 5.00000 \)
- \( \frac{36}{7} \approx 5.14286 \)
5. **Selecting a suitable fraction**:
All fractions from \( \frac{31}{7} \) to \( \frac{35}{7} \) are between \( 4.35425 \) and \( 5.17157 \).
We can choose any of these fractions. To provide one example:
\[
\frac{34}{7}
\]
Thus, the answer is:
\[
\boxed{\frac{34}{7}}
\]
Answered by
GPT-4o mini
Sure! A rational number with a denominator of 7 that is between 7 minus the square root of 7 and 8 minus the square root of 8 is 34 over 7.
So the answer is:
34/7.
So the answer is:
34/7.
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