Answer :
Sure! Let's determine which numbers are irrational from the given options.
To begin, recall that an irrational number is a number that cannot be expressed as a simple fraction (ratio of two integers) and has a non-repeating, non-terminating decimal expansion.
We have four square roots to evaluate:
1. [tex]\(\sqrt{64}\)[/tex]
2. [tex]\(\sqrt{47}\)[/tex]
3. [tex]\(\sqrt{169}\)[/tex]
4. [tex]\(\sqrt{26}\)[/tex]
### Evaluating the Numbers
1. [tex]\(\sqrt{64}\)[/tex]:
[tex]\[
\sqrt{64} = 8
\][/tex]
Since 8 is an integer, [tex]\(\sqrt{64}\)[/tex] is a rational number.
2. [tex]\(\sqrt{47}\)[/tex]:
[tex]\[
\sqrt{47} \approx 6.8556546\ldots
\][/tex]
The square root of 47 is not a perfect square and its decimal expansion is non-repeating and non-terminating. Hence, [tex]\(\sqrt{47}\)[/tex] is an irrational number.
3. [tex]\(\sqrt{169}\)[/tex]:
[tex]\[
\sqrt{169} = 13
\][/tex]
Since 13 is an integer, [tex]\(\sqrt{169}\)[/tex] is a rational number.
4. [tex]\(\sqrt{26}\)[/tex]:
[tex]\[
\sqrt{26} \approx 5.0990195\ldots
\][/tex]
The square root of 26 is not a perfect square and its decimal expansion is non-repeating and non-terminating. Hence, [tex]\(\sqrt{26}\)[/tex] is an irrational number.
### Conclusion
By examining each number, we found that:
- [tex]\(\sqrt{47}\)[/tex] is irrational.
- [tex]\(\sqrt{26}\)[/tex] is irrational.
Therefore, the numbers that are irrational are:
- A. [tex]\(\sqrt{47}\)[/tex]
- C. [tex]\(\sqrt{26}\)[/tex]
To begin, recall that an irrational number is a number that cannot be expressed as a simple fraction (ratio of two integers) and has a non-repeating, non-terminating decimal expansion.
We have four square roots to evaluate:
1. [tex]\(\sqrt{64}\)[/tex]
2. [tex]\(\sqrt{47}\)[/tex]
3. [tex]\(\sqrt{169}\)[/tex]
4. [tex]\(\sqrt{26}\)[/tex]
### Evaluating the Numbers
1. [tex]\(\sqrt{64}\)[/tex]:
[tex]\[
\sqrt{64} = 8
\][/tex]
Since 8 is an integer, [tex]\(\sqrt{64}\)[/tex] is a rational number.
2. [tex]\(\sqrt{47}\)[/tex]:
[tex]\[
\sqrt{47} \approx 6.8556546\ldots
\][/tex]
The square root of 47 is not a perfect square and its decimal expansion is non-repeating and non-terminating. Hence, [tex]\(\sqrt{47}\)[/tex] is an irrational number.
3. [tex]\(\sqrt{169}\)[/tex]:
[tex]\[
\sqrt{169} = 13
\][/tex]
Since 13 is an integer, [tex]\(\sqrt{169}\)[/tex] is a rational number.
4. [tex]\(\sqrt{26}\)[/tex]:
[tex]\[
\sqrt{26} \approx 5.0990195\ldots
\][/tex]
The square root of 26 is not a perfect square and its decimal expansion is non-repeating and non-terminating. Hence, [tex]\(\sqrt{26}\)[/tex] is an irrational number.
### Conclusion
By examining each number, we found that:
- [tex]\(\sqrt{47}\)[/tex] is irrational.
- [tex]\(\sqrt{26}\)[/tex] is irrational.
Therefore, the numbers that are irrational are:
- A. [tex]\(\sqrt{47}\)[/tex]
- C. [tex]\(\sqrt{26}\)[/tex]
