Sure, let's determine which of the given numbers are irrational.
1. [tex]\(\frac{9}{15}\)[/tex]:
- Simplify [tex]\(\frac{9}{15}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 3.
[tex]\[
\frac{9}{15} = \frac{9 \div 3}{15 \div 3} = \frac{3}{5}
\][/tex]
- [tex]\(\frac{3}{5}\)[/tex] is a ratio of two integers, so it is a rational number.
2. [tex]\(\sqrt{18}\)[/tex]:
- The square root of 18 is not a perfect square, since there is no integer [tex]\( n \)[/tex] such that [tex]\( n^2 = 18 \)[/tex].
- Therefore, [tex]\(\sqrt{18}\)[/tex] is an irrational number.
3. [tex]\(\sqrt{9}\)[/tex]:
- The square root of 9 is a perfect square, as [tex]\( 3^2 = 9 \)[/tex].
- Therefore, [tex]\(\sqrt{9} = 3\)[/tex], which is an integer, making it a rational number.
4. [tex]\(\sqrt{169}\)[/tex]:
- The square root of 169 is a perfect square, as [tex]\( 13^2 = 169 \)[/tex].
- Therefore, [tex]\(\sqrt{169} = 13\)[/tex], which is an integer, making it a rational number.
5. [tex]\(\sqrt{78}\)[/tex]:
- The square root of 78 is not a perfect square, since there is no integer [tex]\( n \)[/tex] such that [tex]\( n^2 = 78 \)[/tex].
- Therefore, [tex]\(\sqrt{78}\)[/tex] is an irrational number.
6. [tex]\(\pi\)[/tex]:
- The number [tex]\(\pi\)[/tex], which is the ratio of a circle's circumference to its diameter, is known to be a non-repeating, non-terminating decimal.
- Therefore, [tex]\(\pi\)[/tex] is an irrational number.
To summarize, the irrational numbers among the given options are:
[tex]\[
\sqrt{18}, \sqrt{78}, \pi
\][/tex]
