Let's analyze the given expressions to determine why they are not monomials and then identify which expressions are monomials from the second list.
### Expressions That Are Not Monomials:
1. [tex]\(3cd^X\)[/tex]:
- A monomial is a single term algebraic expression that consists of a constant multiplied by variables raised to non-negative integer powers.
- The variable [tex]\(X\)[/tex] in the exponent makes the expression non-eligible as a monomial because [tex]\(X\)[/tex] can be any number, and we require exponents to be non-negative integers.
2. [tex]\(x + 2w\)[/tex]:
- This expression consists of two terms, [tex]\(x\)[/tex] and [tex]\(2w\)[/tex], separated by a plus sign.
- A monomial must be a single term only.
3. [tex]\(\frac{3}{h}\)[/tex]:
- A monomial cannot have a variable in the denominator as this implies a negative exponent (e.g., [tex]\(\frac{3}{h} = 3h^{-1}\)[/tex]).
- Monomials must have variables with non-negative integer exponents.
4. [tex]\(ab^{-1}\)[/tex]:
- The expression [tex]\(ab^{-1}\)[/tex] includes a variable with a negative exponent ([tex]\(b^{-1}\)[/tex]).
- Exponents must be non-negative integers for an expression to be a monomial.
### Expressions That Are Monomials:
1. [tex]\(-4 + 6\)[/tex]:
- This expression simplifies to [tex]\(2\)[/tex], which is a constant and can be considered a monomial (a constant is a special case of a monomial).
2. [tex]\(b + 2b + 2\)[/tex]:
- This simplifies to [tex]\(3b + 2\)[/tex], which is a binomial, not a monomial, because it consists of two terms.
3. [tex]\((x - 2x)^2\)[/tex]:
- Simplifying inside the parentheses: [tex]\(x - 2x = -x\)[/tex], and then [tex]\((-x)^2 = x^2\)[/tex].
- This is indeed a monomial as it is a single term with a non-negative integer exponent.
4. [tex]\(\frac{rs}{t}\)[/tex]:
- As a fraction with a variable in the denominator, this implies a negative exponent ([tex]\(rst^{-1}\)[/tex]).
- This cannot be a monomial due to the negative exponent.
5. [tex]\(36x^2yz^3\)[/tex]:
- This is a single term expression with each variable having a non-negative integer exponent.
- It satisfies the definition of a monomial.
6. [tex]\(a^x\)[/tex]:
- The variable [tex]\(x\)[/tex] as the exponent makes this expression ineligible to be called a monomial since the exponents must be non-negative integers.
### Summary:
- Expressions that are monomials: [tex]\(-4 + 6\)[/tex] and [tex]\(36x^2yz^3\)[/tex].
- Expressions that are not monomials: [tex]\(3cd^X\)[/tex], [tex]\(x + 2w\)[/tex], [tex]\(\frac{3}{h}\)[/tex], [tex]\(ab^{-1}\)[/tex], [tex]\(b + 2b + 2\)[/tex], [tex]\(\frac{rs}{t}\)[/tex], and [tex]\(a^x\)[/tex].
