Show an example of a rational number that is between square root of 10 and square root of 11
An example of a rational number between √10 and √11 is √(10.5), which is approximately 3.24.
To find a rational number between √10 and √11, we can use the fact that rational numbers can be expressed as fractions.
To start, we need to determine a common denominator for the two square roots. In this case, we'll choose 100, since it's a multiple of both 10 and 11.
√10 can be approximated as √(10/1) × √(100/100), which equals √(1000/100). Simplifying, we get √(1000) / √(100), which is 10√10 / 10.
Similarly, √11 can be approximated as √(11/1) × √(100/100), which equals √(1100/100). Simplifying, we get √(1100) / √(100), which is 10√11 / 10.
Now we have two rational numbers: 10√10 / 10 and 10√11 / 10. To find a number between them, we'll calculate the average of the two.
Average = (10√10 / 10 + 10√11 / 10) / 2
Simplifying further,
Average = (√10 + √11) / 2
Therefore, the rational number between √10 and √11 is (√10 + √11) / 2.