Show an example of a rational number that is between 10−−√ and 11−−√.(1 point)%0D%0AResponses%0D%0A%0D%0A165%0D%0AStart Fraction 16 over 5 end fraction%0D%0A%0D%0A3.1%0D%0A3.1%0D%0A%0D%0A103%0D%0AStart Fraction 10 over 3 end fraction%0D%0A%0D%0A3.4

Question

4. Which irrational number is between 14−−√ and 18−−√ ?(1 point)%0D%0AResponses%0D%0A%0D%0A13−−√%0D%0Astart root 13 end root%0D%0A%0D%0A15−−√%0D%0Astart root 15 end root%0D%0A%0D%0A19−−√%0D%0Astart root 19 end root%0D%0A%0D%0A10−−√

How many irrational numbers are there between 11−−√ and 17−−√?(1 point)%0D%0AResponses%0D%0A%0D%0Afour numbers%0D%0Afour numbers%0D%0A%0D%0Atwenty-eight numbers%0D%0Atwenty-eight numbers%0D%0A%0D%0Aan infinite number%0D%0Aan infinite number%0D%0A%0D%0Asix numbers

Answer 1

Let's tackle each question one by one.

1. Example of a rational number between \( \sqrt{10} \) and \( \sqrt{11} \):

   • We know \( \sqrt{10} \approx 3.16 \) and \( \sqrt{11} \approx 3.32 \).
   • A rational number in this range could be \( \frac{16}{5} = 3.2 \), which is between \( \sqrt{10} \) and \( \sqrt{11} \).
   • Thus, one example is \( \frac{16}{5} \).

2. Irrational number between \( \sqrt{14} \) and \( \sqrt{18} \):

   • We approximate \( \sqrt{14} \approx 3.74 \) and \( \sqrt{18} \approx 4.24 \).
   • Among the options:
     • \( \sqrt{15} \approx 3.87 \) is between those two values.
   • Therefore, the irrational number between \( \sqrt{14} \) and \( \sqrt{18} \) is \( \sqrt{15} \).

3. How many irrational numbers are there between \( \sqrt{11} \) and \( \sqrt{17} \):

   • We calculate \( \sqrt{11} \approx 3.32 \) and \( \sqrt{17} \approx 4.12 \).
   • There are an infinite number of irrational numbers between any two distinct real numbers.
   • Thus, the answer is an infinite number.

In summary:

1. \( \frac{16}{5} \) is a rational number between \( \sqrt{10} \) and \( \sqrt{11} \).
2. \( \sqrt{15} \) is an irrational number between \( \sqrt{14} \) and \( \sqrt{18} \).
3. There are an infinite number of irrational numbers between \( \sqrt{11} \) and \( \sqrt{17} \).

Answer 2

1. Show an example of an irrational number between 9 and 11.
99 squared

2. What is the least integer whose square root is an irrational number between 5 and 7?
26

3. Show an example of a rational number that is between 10 squared and 11 squared.
16/5

4. Which irrational number is between 14 squared and 18 squared?
15 squared

5. How many irrational numbers are there between 11 squared and 17 squared?
an infinite number.

ALL ANSWERS

Answer 3

Let's analyze each of your questions and verify the answers:

1. Show an example of an irrational number between 9 and 11:

   • The square roots of 9 and 11 are \( 3 \) and \( \sqrt{11} \approx 3.32 \) respectively.
   • An example of an irrational number between \( 9 \) and \( 11 \) could be \( \sqrt{10} \) (approximately 3.16).
   • However, if you meant between \( 9 \) and \( 11 \) as actual numbers, \( \sqrt{10} \) (which is approximately 3.16) would work. But if the answer is referring to squaring, then \( 99 \) squared (or \( 9.9^2 = 98.01 \)) is valid because it is still between 81 (9^2) and 121 (11^2).

2. What is the least integer whose square root is an irrational number between 5 and 7?

   • The least integer is indeed \( 26 \) because \( \sqrt{25} = 5 \) and \( \sqrt{36} = 6 \). The integer \( 26 \) gives \( \sqrt{26} \) which is approximately \( 5.1 \), thus an irrational number between 5 and 6.

3. Show an example of a rational number that is between \( 10^2 \) and \( 11^2 \):

   • \( 10^2 = 100 \) and \( 11^2 = 121 \).
   • An example of a rational number in this range is \( \frac{116}{5} = 23.2 \), or simply \( 110 \). The answer given as \( \frac{16}{5} \) (which equals 3.2) does not fit the range between 100 and 121.

4. Which irrational number is between \( 14^2 \) and \( 18^2 \)?

   • \( 14^2 = 196 \) and \( 18^2 = 324 \).
   • An irrational number between those values could be \( \sqrt{200} \) (approximately 14.14), or \( \sqrt{250} \). If you meant \( 15^2 = 225 \), while that is a number in the range, it’s actually rational, so an example isn’t \( 15^2 \) itself but possibly an irrational number such as \( \sqrt{210} \).

5. How many irrational numbers are there between \( 11^2 \) and \( 17^2 \)?

   • \( 11^2 = 121 \) and \( 17^2 = 289 \). There are indeed an infinite number of irrational numbers between these two values.

Summary of your answers with clarifications:

1. Irrational number between 9 and 11: \( \sqrt{10} \) is a good example.
2. Least integer with an irrational square root between 5 and 7: Correct (26).
3. Rational number between \( 10^2 \) and \( 11^2 \): \( 110 \) is better; \( \frac{16}{5} \) doesn’t fit the range.
4. Irrational number between \( 14^2 \) and \( 18^2 \): A better answer is \( \sqrt{210} \) or similar.
5. Irrational numbers between \( 11^2 \) and \( 17^2 \): Correct (an infinite number).

Feel free to ask if you have any further questions or need clarifications!