To determine which number is irrational among the given options, let’s evaluate each one step by step:
1. First number: [tex]\(\frac{\sqrt[3]{-27}}{3}\)[/tex]
- Evaluate the cube root of -27:
[tex]\[
\sqrt[3]{-27} = -3
\][/tex]
- Divide by 3:
[tex]\[
\frac{-3}{3} = -1
\][/tex]
- [tex]\(-1\)[/tex] is a rational number because it can be expressed as the ratio of two integers.
2. Second number: [tex]\(\frac{25}{9}\)[/tex]
- The fraction [tex]\(\frac{25}{9}\)[/tex] is already in the form of a ratio of two integers.
- Therefore, [tex]\(\frac{25}{9}\)[/tex] is a rational number.
3. Third number: [tex]\(\frac{\pi}{\sqrt[3]{27}}\)[/tex]
- Evaluate the cube root of 27:
[tex]\[
\sqrt[3]{27} = 3
\][/tex]
- Divide [tex]\(\pi\)[/tex] by 3:
[tex]\[
\frac{\pi}{3}
\][/tex]
- [tex]\(\pi\)[/tex] is an irrational number, and performing any arithmetic operation (other than multiplication/division by 1) on an irrational number typically results in another irrational number. Thus, [tex]\(\frac{\pi}{3}\)[/tex] is an irrational number.
4. Fourth number: [tex]\(\overline{12}\)[/tex] (the repeating decimal notation for [tex]\(12.121212...\)[/tex])
- A repeating decimal can be expressed as a fraction. For example, if we let [tex]\(x = 12.121212...\)[/tex], then:
[tex]\[
100x = 1212.1212...
\][/tex]
- Subtracting [tex]\(x\)[/tex] from [tex]\(100x\)[/tex]:
[tex]\[
100x - x = 1212.1212... - 12.1212... \implies 99x = 1200 \implies x = \frac{1200}{99}
\][/tex]
- Being able to express the repeating decimal as a fraction means that [tex]\(\overline{12}\)[/tex] is a rational number.
Conclusion:
Among the options provided, the third number [tex]\(\frac{\pi}{\sqrt[3]{27}} = \frac{\pi}{3}\)[/tex] is the only irrational number.
