Question
what rational number with the denominator of 7 is between √8 and √9
Answers
Answer
ok can you make it a bit more less of work
Answered by
oobleck
to make it even easier, consider that √9 = 3. so just subtract 1/7 from that, giving 2 6/7 = 20/7
Answered by
GPT-4o mini
To find a rational number with a denominator of 7 that is between \( \sqrt{8} \) and \( \sqrt{9} \), we first calculate the values of these square roots:
\[
\sqrt{8} \approx 2.828
\]
\[
\sqrt{9} = 3
\]
Now we are looking for a rational number in the form \( \frac{n}{7} \) such that:
\[
2.828 < \frac{n}{7} < 3
\]
To find the corresponding values for \( n \), we can multiply the inequalities by 7:
\[
7 \times 2.828 < n < 7 \times 3
\]
Calculating these products:
\[
19.796 < n < 21
\]
Since \( n \) must be an integer, the possible integer value for \( n \) is 20.
Now, we can check this value:
\[
\frac{20}{7} \approx 2.857
\]
Since \( 2.828 < 2.857 < 3 \), we find that \( \frac{20}{7} \) is indeed between \( \sqrt{8} \) and \( \sqrt{9} \).
Thus, the rational number with the denominator of 7 that is between \( \sqrt{8} \) and \( \sqrt{9} \) is:
\[
\frac{20}{7}
\]
\[
\sqrt{8} \approx 2.828
\]
\[
\sqrt{9} = 3
\]
Now we are looking for a rational number in the form \( \frac{n}{7} \) such that:
\[
2.828 < \frac{n}{7} < 3
\]
To find the corresponding values for \( n \), we can multiply the inequalities by 7:
\[
7 \times 2.828 < n < 7 \times 3
\]
Calculating these products:
\[
19.796 < n < 21
\]
Since \( n \) must be an integer, the possible integer value for \( n \) is 20.
Now, we can check this value:
\[
\frac{20}{7} \approx 2.857
\]
Since \( 2.828 < 2.857 < 3 \), we find that \( \frac{20}{7} \) is indeed between \( \sqrt{8} \) and \( \sqrt{9} \).
Thus, the rational number with the denominator of 7 that is between \( \sqrt{8} \) and \( \sqrt{9} \) is:
\[
\frac{20}{7}
\]
Answered by
GPT-4o mini
Sure! To find a rational number with a denominator of 7 that falls between \( \sqrt{8} \) and \( \sqrt{9} \):
1. We know \( \sqrt{8} \approx 2.828 \) and \( \sqrt{9} = 3 \).
2. We need \( \frac{n}{7} \) such that \( 2.828 < \frac{n}{7} < 3 \).
3. Thus, we multiply by 7 to find \( n \):
- Multiply \( 2.828 \) by 7: approximately \( 19.796 \).
- Multiply \( 3 \) by 7: \( 21 \).
4. The integer \( n \) must be \( 20 \) (since it's the only integer between \( 19.796 \) and \( 21 \)).
Therefore, the rational number is:
\[
\frac{20}{7}
\]
This is your answer in a more concise format!
1. We know \( \sqrt{8} \approx 2.828 \) and \( \sqrt{9} = 3 \).
2. We need \( \frac{n}{7} \) such that \( 2.828 < \frac{n}{7} < 3 \).
3. Thus, we multiply by 7 to find \( n \):
- Multiply \( 2.828 \) by 7: approximately \( 19.796 \).
- Multiply \( 3 \) by 7: \( 21 \).
4. The integer \( n \) must be \( 20 \) (since it's the only integer between \( 19.796 \) and \( 21 \)).
Therefore, the rational number is:
\[
\frac{20}{7}
\]
This is your answer in a more concise format!
Answered by
GPT-4o mini
You're absolutely right!
Starting with \( \sqrt{9} = 3 \), you can simply subtract \( \frac{1}{7} \) to find a rational number just below it:
\[
3 - \frac{1}{7} = \frac{21}{7} - \frac{1}{7} = \frac{20}{7}
\]
So, \( \frac{20}{7} \) is indeed a rational number between \( \sqrt{8} \) and \( \sqrt{9} \).
Thank you for pointing out the simpler method!
Starting with \( \sqrt{9} = 3 \), you can simply subtract \( \frac{1}{7} \) to find a rational number just below it:
\[
3 - \frac{1}{7} = \frac{21}{7} - \frac{1}{7} = \frac{20}{7}
\]
So, \( \frac{20}{7} \) is indeed a rational number between \( \sqrt{8} \) and \( \sqrt{9} \).
Thank you for pointing out the simpler method!