To determine which of the given numbers is not a rational number, let's analyze each number individually.
1. [tex]\(-5 + \frac{4}{11}\)[/tex]:
- [tex]\( -5 + \frac{4}{11} = -5 + 0.363636...\)[/tex].
- This can be written as the fraction [tex]\(-\frac{55}{11} + \frac{4}{11} = -\frac{51}{11}\)[/tex], which is a rational number because it can be expressed as a ratio of two integers. Thus, this number is rational.
2. [tex]\(\sqrt{31}\)[/tex]:
- The square root of 31 ([tex]\(\sqrt{31}\)[/tex]) is an irrational number because 31 is not a perfect square, meaning its square root cannot be expressed as a fraction of two integers.
3. 7.608:
- The number 7.608 can be written as the fraction [tex]\(\frac{7608}{1000}\)[/tex] by moving the decimal point three places to the right. Simplifying this fraction gives [tex]\(\frac{3804}{500} = \frac{1902}{250} = \frac{951}{125}\)[/tex], which confirms it is a rational number.
4. [tex]\(18.4 \overline{\overline{6}}\)[/tex]:
- The number [tex]\(18.4 \overline{\overline{6}}\)[/tex] (meaning [tex]\(18.466666...\)[/tex] with the '6' repeating indefinitely) can be written as the sum of a terminating decimal and a repeating fraction: [tex]\(18.4 + 0.066666...\)[/tex].
- The repeating part [tex]\(0.066666...\)[/tex] can be expressed as [tex]\(\frac{2}{30} = \frac{1}{15}\)[/tex], and thus [tex]\( \infty\)[/tex]number [tex]\(18.4 \overline{\overline{6}}\)[/tex] is rational.
In conclusion, among the given numbers, the one that is not a rational number is [tex]\(\sqrt{31}\)[/tex].
