Let's determine which of the given expressions are monomials. A monomial is a single term that is a product of constants and/or variables raised to non-negative integer powers.
1. [tex]$-7$[/tex]
Analysis: This is a constant term. A constant is a special case of a monomial where the variable exponent is zero (e.g., [tex]$-7$[/tex] can be seen as [tex]$-7 \cdot x^0$[/tex]).
Conclusion: Yes, [tex]$-7$[/tex] is a monomial.
2. a
Analysis: This is a single variable. Variables raised to the power of 1 are considered monomials.
Conclusion: Yes, [tex]\( a \)[/tex] is a monomial.
3. [tex]$x + y$[/tex]
Analysis: This is a sum of two terms. Monomials cannot contain addition or subtraction. They should be a single term.
Conclusion: No, [tex]$x + y$[/tex] is not a monomial.
4. [tex]$\frac{1}{x}$[/tex]
Analysis: This expression involves a variable in the denominator, which can be written as [tex]$x^{-1}$[/tex]. Monomials require the variables to have non-negative integer exponents.
Conclusion: No, [tex]$\frac{1}{x}$[/tex] is not a monomial.
5. [tex]$24r^2st^3$[/tex]
Analysis: This is a single term composed of a constant (24) and variables raised to non-negative integer powers ([tex]$r^2, s^1, t^3$[/tex]).
Conclusion: Yes, [tex]$24r^2st^3$[/tex] is a monomial.
6. [tex]$\frac{ab}{5}$[/tex]
Analysis: This expression involves a division by 5, which we can rewrite as [tex]$(1/5)ab$[/tex]. While the fractional coefficient alone is allowed, we need to ensure the variables meet the criteria: [tex]$a$[/tex] and [tex]$b$[/tex] are each raised to the power of 1, but the fractional coefficient does not fit the form normally required for a monomial.
Conclusion: No, [tex]$\frac{ab}{5}$[/tex] is not a monomial.
7. [tex]$b^x$[/tex]
Analysis: This expression contains a variable ([tex]$b$[/tex]) raised to another variable ([tex]$x$[/tex]). Monomials require the exponents to be non-negative integers, not variables.
Conclusion: No, [tex]$b^x$[/tex] is not a monomial.
Summary of Monomials:
- [tex]$-7$[/tex]: Yes
- [tex]$a$[/tex]: Yes
- [tex]$x + y$[/tex]: No
- [tex]$\frac{1}{x}$[/tex]: No
- [tex]$24r^2st^3$[/tex]: Yes
- [tex]$\frac{ab}{5}$[/tex]: No
- [tex]$b^x$[/tex]: No
