Classroom Management:
801 MATH ROCKS

Task: MathXL for School: Practice & Problem Solving

Which numbers are irrational?

\[ \sqrt{64} \]
\[ \sqrt{47} \]
\[ \sqrt{169} \]
\[ \sqrt{26} \]

Which numbers are irrational? Select all that apply.
A. \[ \sqrt{47} \]
B. \[ \sqrt{64} \]
C. \[ \sqrt{26} \]
D. \[ \sqrt{169} \]

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]