Question
Which numbers are irrational? Select the three that apply.(3 points)Responses9159 over 159–√square root of 918−−√square root of 18πpi169−−−√square root of 16978−−√
Answers
GPT-4o mini
To determine which numbers from the provided options are irrational, we need to identify whether each number can be expressed as a fraction of two integers or is a non-repeating, non-terminating decimal.
Let's go through the options:
1. **\( \frac{9159}{159} \)**: This is a fraction of two integers, so it is rational.
2. **\( \sqrt{9} \)**: The square root of 9 is 3, which is rational.
3. **\( \sqrt{18} \)**: This number can be simplified to \( 3\sqrt{2} \). The square root of 2 is irrational, so \( \sqrt{18} \) is irrational.
4. **\( \pi \)**: Pi is a well-known irrational number.
5. **\( \sqrt{169} \)**: The square root of 169 is 13, which is rational.
6. **\( 78 \)**: This is an integer and can be expressed as \( \frac{78}{1} \), so it is rational.
From this analysis, the three irrational numbers are:
- \( \sqrt{18} \)
- \( \pi \)
The third option listed as irrational might be a misinterpretation since only two irrational numbers are presented in the provided options. If you are looking for three selections and \( \sqrt{18} \) and \( \pi \) must be included, I would need additional context to identify a third irrational number, as none of the other choices fit that criterion.
So to summarize:
- \( \sqrt{18} \)
- \( \pi \)
And from the given numbers, only these are irrational.
Let's go through the options:
1. **\( \frac{9159}{159} \)**: This is a fraction of two integers, so it is rational.
2. **\( \sqrt{9} \)**: The square root of 9 is 3, which is rational.
3. **\( \sqrt{18} \)**: This number can be simplified to \( 3\sqrt{2} \). The square root of 2 is irrational, so \( \sqrt{18} \) is irrational.
4. **\( \pi \)**: Pi is a well-known irrational number.
5. **\( \sqrt{169} \)**: The square root of 169 is 13, which is rational.
6. **\( 78 \)**: This is an integer and can be expressed as \( \frac{78}{1} \), so it is rational.
From this analysis, the three irrational numbers are:
- \( \sqrt{18} \)
- \( \pi \)
The third option listed as irrational might be a misinterpretation since only two irrational numbers are presented in the provided options. If you are looking for three selections and \( \sqrt{18} \) and \( \pi \) must be included, I would need additional context to identify a third irrational number, as none of the other choices fit that criterion.
So to summarize:
- \( \sqrt{18} \)
- \( \pi \)
And from the given numbers, only these are irrational.