Let's evaluate whether each of the given expressions is a monomial. A monomial is defined as a single term algebraic expression that consists of a coefficient, variables, and their exponents. It should not include addition, subtraction, or division by variables. Keeping these criteria in mind, let's analyze each expression:
1. [tex]\(-7\)[/tex]
- This is a single term consisting of just a constant. No variables or operations are involved.
- Monomial: YES
2. [tex]\(a\)[/tex]
- This is a single variable with an implied exponent of [tex]\(1\)[/tex]. No operations or additional terms are involved.
- Monomial: YES
3. [tex]\(x + y\)[/tex]
- This expression consists of two terms combined by addition, which disqualifies it as a monomial.
- Monomial: NO
4. [tex]\(\frac{1}{x}\)[/tex]
- This expression involves division by a variable ([tex]\(x\)[/tex]), which is not allowed in a monomial.
- Monomial: NO
5. [tex]\(24 r^2 s t^3\)[/tex]
- This is a single term with a constant coefficient [tex]\(24\)[/tex] and variables [tex]\(r\)[/tex], [tex]\(s\)[/tex], and [tex]\(t\)[/tex] raised to positive integer powers. All parts of this expression fit the definition of a monomial.
- Monomial: YES
6. [tex]\(\frac{a b}{5}\)[/tex]
- Although this single term involves multiplication of variables [tex]\(a\)[/tex] and [tex]\(b\)[/tex], it includes division by a constant [tex]\(5\)[/tex]. Since division by a constant is allowed, it can still be considered a single term.
- Monomial: YES
7. [tex]\(b^x\)[/tex]
- This expression has a variable ([tex]\(b\)[/tex]) raised to the power of another variable ([tex]\(x\)[/tex]), which disqualifies it from being a monomial.
- Monomial: NO
In conclusion, the expressions that are monomials are:
- [tex]\(-7\)[/tex]
- [tex]\(a\)[/tex]
- [tex]\(24 r^2 s t^3\)[/tex]
- [tex]\(\frac{a b}{5}\)[/tex]
Therefore, the monomials among the given expressions are:
[tex]\[
\boxed{-7, \, a, \, 24 r^2 s t^3, \, \frac{a b}{5}}
\][/tex]
