Show an example of a rational number that is between 10−−√ and 11−−√.
To find a rational number that is between √10 and √11, we need to find two perfect square numbers that √10 and √11 lie between.
√10 is between √9 and √16, so we can use the rational number 3/2 which is between 9/4 and 16/4.
√11 is between √9 and √16, so we can use the rational number 4/3 which is between 9/4 and 16/4.
Therefore, a rational number that is between √10 and √11 is 3/2.
Show an example of a rational number that is between √10 and √11.
a. 3.4
b. 16/5
c. 3.1
d. 10/3
To find a rational number between √10 and √11, we can take the average of these two numbers.
Step 1: Find the average of √10 and √11.
Average = (√10 + √11) / 2
Step 2: Simplify the expression.
Average = (√10 + √11) / 2 = (√10 + √11) / 2 * (√10 - √11) / (√10 - √11)
Average = (√10 * √10 - √11 * √10 + √10 * √11 - √11 * √11) / (2 * (√10 - √11))
Average = (10 - √110 + √110 - 11) / (2 * (√10 - √11))
Average = (10 - 11) / (2 * (√10 - √11))
Average = -1 / (2 * (√10 - √11))
So, a rational number between √10 and √11 is -1 / (2 * (√10 - √11)).
To find a rational number between two given irrational numbers, we can rationalize the denominators and choose a suitable numerator.
Given that the numbers are √10 and √11, let's start by rationalizing their denominators:
√10 = (√10 * √10) / √10 = 10 / √10
√11 = (√11 * √11) / √11 = 11 / √11
Now, to find a rational number between these two, we need to find a rational number between 10 / √10 and 11 / √11.
To do this, let's find a common denominator. The least common multiple (LCM) of √10 and √11 is √110, so we can rewrite the expression as:
(10 / √10) * (√110 / √110) = (10 * √110) / (√10 * √110) = (10√110) / √110 = 10√110 / 110 = √(1000/11)
Hence, a rational number between √10 and √11 is √(1000/11), which simplifies to approximately 9.4868.
Therefore, an example of a rational number between √10 and √11 is 9.4868.