To determine which number produces an irrational number when multiplied by [tex]\(\frac{1}{3}\)[/tex], we'll examine each option step-by-step:
### Option A: [tex]\(\frac{2}{3}\)[/tex]
Multiply [tex]\(\frac{2}{3}\)[/tex] by [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\frac{2}{3} \times \frac{1}{3} = \frac{2}{9}
\][/tex]
[tex]\(\frac{2}{9}\)[/tex] is a rational number because it can be expressed as the ratio of two integers (2 and 9).
### Option B: [tex]\(-\sqrt{17}\)[/tex]
Multiply [tex]\(-\sqrt{17}\)[/tex] by [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
-\sqrt{17} \times \frac{1}{3} = -\frac{\sqrt{17}}{3}
\][/tex]
[tex]\(\sqrt{17}\)[/tex] is an irrational number, and multiplying an irrational number by a non-zero rational number ([tex]\(\frac{1}{3}\)[/tex]) results in an irrational number. Therefore, [tex]\(-\frac{\sqrt{17}}{3}\)[/tex] is irrational.
### Option C: 0.166
Multiply 0.166 by [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
0.166 \times \frac{1}{3} \approx 0.0553333
\][/tex]
Since [tex]\(0.166\)[/tex] has a finite decimal expansion, it represents a rational number. Its product with another rational number ([tex]\(\frac{1}{3}\)[/tex]) is also rational (specifically [tex]\(\frac{83}{1500}\)[/tex], which can be simplified but remains rational).
### Option D: 2
Multiply 2 by [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
2 \times \frac{1}{3} = \frac{2}{3}
\][/tex]
[tex]\(\frac{2}{3}\)[/tex] is clearly a rational number because it can be expressed as the ratio of two integers (2 and 3).
### Conclusion
The number that produces an irrational number when multiplied by [tex]\(\frac{1}{3}\)[/tex] is [tex]\(\boxed{-\sqrt{17}}\)[/tex].
