Question

Which irrational number is between 14−−√ and 18−−√ ?(1 point)Responses13−−√start root 13 end root10−−√start root 10 end root15−−√start root 15 end root19−−√

Answer 1

To determine which irrational number is between \( \sqrt{14} \) and \( \sqrt{18} \), we start by estimating the square roots:

1. **Estimating \( \sqrt{14} \)**:
- \( \sqrt{16} = 4 \) (since \( 4^2 = 16 \))
- \( \sqrt{14} \) is between \( \sqrt{9} = 3 \) and \( \sqrt{16} = 4 \).
- Let's roughly estimate it:
- \( 3.7^2 = 13.69 \) (too low)
- \( 3.8^2 = 14.44 \) (too high)
- So, \( \sqrt{14} \) is about \( 3.74 \).

2. **Estimating \( \sqrt{18} \)**:
- \( \sqrt{16} = 4 \) and \( \sqrt{25} = 5 \).
- Therefore, \( \sqrt{18} \) is between \( 4 \) and \( 5 \).
- Let's estimate:
- \( 4.2^2 = 17.64 \) (too low)
- \( 4.3^2 = 18.49 \) (too high)
- So, \( \sqrt{18} \) is about \( 4.24 \).

So, we find that \( \sqrt{14} \approx 3.74 \) and \( \sqrt{18} \approx 4.24 \).

Now we can check which options are between these two values:

- \( \sqrt{13} \approx 3.605 \) (too low)
- \( \sqrt{10} \approx 3.162 \) (too low)
- \( \sqrt{15} \approx 3.873 \) (between \( 3.74 \) and \( 4.24 \))
- \( \sqrt{19} \approx 4.359 \) (too high)

The irrational number that falls between \( \sqrt{14} \) and \( \sqrt{18} \) is \( \sqrt{15} \).

Thus, the answer is:
\[
\sqrt{15}
\]