Let's analyze each expression to understand why they are not considered monomials:
1. [tex]\(3 c d^{ X }\)[/tex]:
- A monomial is a single term consisting of a constant coefficient multiplied by one or more variables, where the variables are raised to non-negative integer exponents.
- In the expression [tex]\(3 c d^{ X }\)[/tex], the exponent of [tex]\(d\)[/tex] is [tex]\(X\)[/tex].
- Since [tex]\(X\)[/tex] is a variable and not a non-negative integer, this makes the expression not a monomial.
- Reason: Variable [tex]\(X\)[/tex] in the exponent makes it not a monomial.
2. [tex]\(x + 2w\)[/tex]:
- A monomial must be a single term.
- The expression [tex]\(x + 2w\)[/tex] consists of two separate terms: [tex]\(x\)[/tex] and [tex]\(2w\)[/tex].
- The presence of addition between these terms means it is not a single term.
- Reason: Addition of terms makes it not a single monomial term.
3. [tex]\(\frac{3}{h}\)[/tex]:
- A monomial can involve multiplication of variables, but not division by a variable.
- The expression [tex]\(\frac{3}{h}\)[/tex] can be rewritten as [tex]\(3 h^{-1}\)[/tex].
- Here, [tex]\(h\)[/tex] is in the denominator, which introduces a negative exponent.
- A monomial cannot have a variable with a negative exponent.
- Reason: Division by a variable makes it not a monomial.
4. [tex]\(a b^{-1}\)[/tex]:
- A monomial should not have any variable with a negative exponent.
- In the expression [tex]\(a b^{-1}\)[/tex], the variable [tex]\(b\)[/tex] has an exponent of [tex]\(-1\)[/tex].
- Since a monomial requires all variable exponents to be non-negative integers, this expression does not qualify.
- Reason: Negative exponent makes it not a monomial.
In summary:
- [tex]\(3 c d^{ X }\)[/tex] is not a monomial due to a variable exponent.
- [tex]\(x + 2w\)[/tex] is not a monomial because it consists of multiple terms.
- [tex]\(\frac{3}{h}\)[/tex] is not a monomial due to division by a variable (negative exponent).
- [tex]\(a b^{-1}\)[/tex] is not a monomial because it includes a negative exponent.
