Which of the following is an irrational number between 5 and 7: 6–√ , 12−−√ , 27−−√ , 36−−√ ?(1 point)

Question

What is the largest integer whose square root is an irrational number between 3 and 4?(1 point)

Find a rational number between 2 and 2 1/4
using the method of adding the numbers together and dividing by 2. Write the answer as a mixed number.(1 point)

Find a rational number with one decimal place that is between 17−−√
and 18−−√
.(1 point)

Which of the following is between π
and 11−−√
?(1 point)
Responses

7–√
Start Root 7 End Root

8–√
Start Root 8 End Root

12−−√
Start Root 12 End Root

10−−√

Answer 1

Let's address each question one by one.

1. Irrational number between 5 and 7

   We need to analyze the given numbers:

   • \( 6 - \sqrt{2} \)
   • \( \sqrt{12} \)
   • \( \sqrt{27} \)
   • \( \sqrt{36} = 6 \) (which is not irrational)

   Let's evaluate the irrational values:

   • \( 6 - \sqrt{2} \) is irrational, and \( \sqrt{2} \approx 1.414 \), so \( 6 - \sqrt{2} \approx 4.586 \) (not between 5 and 7).
   • \( \sqrt{12} \approx 3.464 \) (not between 5 and 7).
   • \( \sqrt{27} \approx 5.196 \) (this is between 5 and 7).
   • \( \sqrt{36} = 6 \) (not irrational).

   Therefore, the answer is \( \sqrt{27} \approx 5.196 \).

2. Largest integer whose square root is an irrational number between 3 and 4

   We want to find an integer \( n \) such that \( 3 < \sqrt{n} < 4 \).

   • Squaring the inequalities gives \( 9 < n < 16 \).

   The largest integer \( n \) in this range is 15.

3. Rational number between 2 and \( 2 \frac{1}{4} \)

   To find a rational number between 2 and \( 2 \frac{1}{4} \), first convert \( 2 \frac{1}{4} \) to an improper fraction:

   • \( 2 \frac{1}{4} = \frac{9}{4} \).

   Now, we can find a rational number by adding 2 and \( 2 \frac{1}{4} \) together and dividing by 2: \[ \text{Average} = \frac{2 + 2 \frac{1}{4}}{2} = \frac{2 + \frac{9}{4}}{2} = \frac{\frac{8}{4} + \frac{9}{4}}{2} = \frac{\frac{17}{4}}{2} = \frac{17}{8} = 2 \frac{1}{8}. \] Therefore, a rational number between 2 and \( 2 \frac{1}{4} \) is \( 2 \frac{1}{8} \).

4. Rational number with one decimal place between \( \sqrt{17} \) and \( \sqrt{18} \)

   First, we compute:

   • \( \sqrt{17} \approx 4.123 \)
   • \( \sqrt{18} \approx 4.243 \)

   A rational number with one decimal place that is between these values could be 4.2.

5. Which of the following is between \( \pi \) and \( \sqrt{11} \)

   Approximate the values:

   • \( \pi \approx 3.141 \)
   • \( \sqrt{11} \approx 3.317 \)

   Analyze the options:

   • \( \sqrt{7} \approx 2.646 \) (not between).
   • \( \sqrt{8} \approx 2.828 \) (not between).
   • \( \sqrt{12} \approx 3.464 \) (not between).
   • \( \sqrt{10} \approx 3.162 \) (this is between \( \pi \) and \( \sqrt{11} \)).

   Therefore, the answer is \( \sqrt{10} \).