To determine which of the given numbers is rational, let's analyze each option step by step.
1. -5:
- A rational number is defined as any number that can be expressed as a fraction [tex]\(\frac{p}{q}\)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers, and [tex]\(q \neq 0\)[/tex].
- The number [tex]\(-5\)[/tex] can be expressed as [tex]\(\frac{-5}{1}\)[/tex], where [tex]\(-5\)[/tex] and [tex]\(1\)[/tex] are both integers.
- Therefore, [tex]\(-5\)[/tex] is a rational number.
2. [tex]\(5\pi\)[/tex]:
- [tex]\(\pi\)[/tex] (pi) is a well-known irrational number, which means it cannot be expressed as a fraction of two integers.
- Multiplying an irrational number (like [tex]\(\pi\)[/tex]) by a rational number (such as 5) results in an irrational number.
- Therefore, [tex]\(5\pi\)[/tex] is irrational.
3. [tex]\(5\sqrt{5}\)[/tex]:
- [tex]\(\sqrt{5}\)[/tex] (the square root of 5) is an irrational number because it cannot be expressed as a fraction of two integers.
- Multiplying an irrational number (like [tex]\(\sqrt{5}\)[/tex]) by a rational number (such as 5) results in an irrational number.
- Therefore, [tex]\(5\sqrt{5}\)[/tex] is irrational.
4. [tex]\(5e\)[/tex]:
- [tex]\(e\)[/tex] (Euler's number) is another well-known irrational number.
- Multiplying an irrational number (like [tex]\(e\)[/tex]) by a rational number (such as 5) results in an irrational number.
- Therefore, [tex]\(5e\)[/tex] is irrational.
By eliminating the irrational numbers from the choices, we find that the only rational number in the list is:
[tex]\(\boxed{-5}\)[/tex]
