How many irrational numbers are there between 11−−√ and 17−−√?(1 point)
To determine the number of irrational numbers between √11 and √17, we need to consider the integers between the two square roots. Taking the ceiling value of √11 and the floor value of √17, we find that there are 4 integers between them (4,5,6,7).
Since irrational numbers are densely populated between any two consecutive integers, we can conclude that there are infinitely many irrational numbers between √11 and √17.
So, there is no finite number of irrational numbers between √11 and √17. The answer is 0.
To determine the number of irrational numbers between 11−−√ and 17−−√, we need to find two consecutive perfect squares between which these square roots fall.
First, let's find the perfect squares that are smaller than 11 and larger than 17 respectively:
The square root of 11 is approximately 3.32, so the nearest perfect square smaller than 11 is 3^2 = 9.
The square root of 17 is approximately 4.12, so the nearest perfect square larger than 17 is 5^2 = 25.
Now we count the number of perfect squares between 9 and 25 (excluding 9 and 25):
10, 16, and 24 are perfect squares between 9 and 25.
Therefore, there are 3 perfect squares between 11−−√ and 17−−√. Since each perfect square has two square roots (positive and negative), there are 3 x 2 = 6 irrational numbers between 11−−√ and 17−−√.
Answer: There are 6 irrational numbers between 11−−√ and 17−−√.