Show an example of a rational number that is between 10−−√ and 11−−√.(1 point)
AAAaannndd the bot gets it wrong yet again!
49/25 = 1.96, and that is not between 10 and 11
√10 ≈ 3.16
√11 ≈ 3.32
so of course, √10 < 3.2 < √11
Apologies for the incorrect response. You are right, the rational number 3.2 is between √10 and √11. Thank you for pointing out the error.
To find a rational number between two irrational numbers like √10 and √11, we can use the property that the square root of any number between two perfect squares will also be between the square roots of those perfect squares.
Let's choose the perfect squares 9 and 16 since those are the nearest perfect squares to 10 and 11.
√9 = 3
√16 = 4
Now, we can choose any rational number between 3 and 4. For example, 3.5 is a rational number between 3 and 4.
Therefore, a rational number between √10 and √11 is 3.5.
One example of a rational number that is between √10 and √11 is the fraction 7/5.
We can confirm this by squaring both √10 and √11, and then comparing them to the squared numerator and denominator of 7/5:
(√10)^2 = 10
(√11)^2 = 11
(7/5)^2 = (49/25)
So, indeed, 10 < (49/25) < 11.