To determine which number among the given options is rational, let's recall some key concepts about rational numbers:
### Rational Numbers
- Rational numbers are numbers that can be expressed as the quotient or fraction [tex]\(\frac{p}{q}\)[/tex] of two integers [tex]\(p\)[/tex] and [tex]\(q\)[/tex], where [tex]\(q \neq 0\)[/tex].
- They include:
- Terminating decimals
- Repeating decimals
Now, we will evaluate each option:
#### Option A: [tex]\(0.36458121 \ldots\)[/tex]
This number appears to be a non-repeating, non-terminating decimal. Non-repeating, non-terminating decimals are classified as irrational numbers because they cannot be expressed as a fraction of two integers.
#### Option B: [tex]\(\pi\)[/tex]
The number [tex]\(\pi\)[/tex] (pi) is a well-known mathematical constant approximately equal to 3.14159, which is also a non-repeating, non-terminating decimal. [tex]\(\pi\)[/tex] is classified as an irrational number.
#### Option C: [tex]\(0.777 \ldots\)[/tex]
This number is a repeating decimal (the digits 7 repeat indefinitely). Any repeating decimal can be expressed as a fraction of two integers. For example:
[tex]\[ 0.777\ldots = \frac{7}{9} \][/tex]
Thus, this is a rational number.
#### Option D: [tex]\(\sqrt{5}\)[/tex]
The square root of 5 ([tex]\(\sqrt{5}\)[/tex]) is not a perfect square, and its decimal representation is non-repeating and non-terminating. Therefore, [tex]\(\sqrt{5}\)[/tex] is an irrational number.
### Conclusion
Among the options given, the number that is rational is:
C. [tex]\(0.777 \ldots\)[/tex]
