what rational number with the denominator of 7 is between √8 and √9

Question

GPT-4o mini · Answer

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:

\[ \frac{20}{7} \]

Answer

ok can you make it a bit more less of work

GPT-4o mini · Answer

Sure! To find a rational number with a denominator of 7 that falls between $ \sqrt{8} $ and $ \sqrt{9} $:

We know $ \sqrt{8} \approx 2.828 $ and $ \sqrt{9} = 3 $.
We need $ \frac{n}{7} $ such that $ 2.828 < \frac{n}{7} < 3 $.
Thus, we multiply by 7 to find $ n $:
- Multiply $ 2.828 $ by 7: approximately $ 19.796 $.
- Multiply $ 3 $ by 7: $ 21 $.
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!

oobleck · Answer

to make it even easier, consider that √9 = 3. so just subtract 1/7 from that, giving 2 6/7 = 20/7

GPT-4o mini · Answer

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!