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Sagot :
Alright, let's go through each of the given mathematical expressions and determine why certain expressions are not monomials, and which ones are monomials.
### Why the expressions are not monomials:
1. [tex]\(3cd^X\)[/tex]:
- This expression is not a monomial because the exponent [tex]\(X\)[/tex] is a variable. For an expression to be a monomial, all exponents should be non-negative integers (constants).
2. [tex]\(x + 2w\)[/tex]:
- This expression is not a monomial because it is a sum of two terms ([tex]\(x\)[/tex] and [tex]\(2w\)[/tex]). A monomial can have only one term and cannot be a sum or difference of terms.
3. [tex]\(\frac{3}{h}\)[/tex]:
- This expression is not a monomial because it involves a variable in the denominator. Monomials cannot be fractions with variables in the denominator; they need to be products of constants and variables with non-negative integer exponents.
4. [tex]\(ab^{-1}\)[/tex]:
- This expression is not a monomial because it includes a negative exponent. In [tex]\(ab^{-1}\)[/tex], [tex]\(b^{-1}\)[/tex] implies a negative exponent for [tex]\(b\)[/tex], which disqualifies it from being a monomial.
### Which expressions are monomials:
1. [tex]\(-4 + 6\)[/tex]:
- This expression simplifies to [tex]\(2\)[/tex] (since [tex]\(-4 + 6 = 2\)[/tex]). The result [tex]\(2\)[/tex] is a constant, which is a monomial.
2. [tex]\(b + 2b + 2\)[/tex]:
- This expression simplifies to [tex]\(3b + 2\)[/tex]. Since [tex]\(3b + 2\)[/tex] is a sum of two terms, it is not a monomial.
3. [tex]\((x - 2x)^2\)[/tex]:
- Firstly, simplify the expression inside the parentheses: [tex]\(x - 2x = -x\)[/tex]. Then, square the result: [tex]\((-x)^2 = x^2\)[/tex]. The result [tex]\(x^2\)[/tex] is a monomial.
4. [tex]\(\frac{rs}{t}\)[/tex]:
- This expression is not a monomial because it involves a fraction with a variable [tex]\(t\)[/tex] in the denominator.
5. [tex]\(36x^2yz^3\)[/tex]:
- This expression is a product of constants and variables, where each variable has a non-negative integer exponent. Hence, [tex]\(36x^2yz^3\)[/tex] is a monomial.
6. [tex]\(a^x\)[/tex]:
- This expression is not a monomial because the exponent [tex]\(x\)[/tex] is a variable. Monomials must have variable exponents that are non-negative integers (constants).
### Conclusion:
The expressions that are monomials are:
1. [tex]\((-4 + 6)\)[/tex] which simplifies to [tex]\(2\)[/tex].
2. [tex]\((x - 2x)^2\)[/tex] which simplifies to [tex]\(x^2\)[/tex].
3. [tex]\(36x^2yz^3\)[/tex].
### Why the expressions are not monomials:
1. [tex]\(3cd^X\)[/tex]:
- This expression is not a monomial because the exponent [tex]\(X\)[/tex] is a variable. For an expression to be a monomial, all exponents should be non-negative integers (constants).
2. [tex]\(x + 2w\)[/tex]:
- This expression is not a monomial because it is a sum of two terms ([tex]\(x\)[/tex] and [tex]\(2w\)[/tex]). A monomial can have only one term and cannot be a sum or difference of terms.
3. [tex]\(\frac{3}{h}\)[/tex]:
- This expression is not a monomial because it involves a variable in the denominator. Monomials cannot be fractions with variables in the denominator; they need to be products of constants and variables with non-negative integer exponents.
4. [tex]\(ab^{-1}\)[/tex]:
- This expression is not a monomial because it includes a negative exponent. In [tex]\(ab^{-1}\)[/tex], [tex]\(b^{-1}\)[/tex] implies a negative exponent for [tex]\(b\)[/tex], which disqualifies it from being a monomial.
### Which expressions are monomials:
1. [tex]\(-4 + 6\)[/tex]:
- This expression simplifies to [tex]\(2\)[/tex] (since [tex]\(-4 + 6 = 2\)[/tex]). The result [tex]\(2\)[/tex] is a constant, which is a monomial.
2. [tex]\(b + 2b + 2\)[/tex]:
- This expression simplifies to [tex]\(3b + 2\)[/tex]. Since [tex]\(3b + 2\)[/tex] is a sum of two terms, it is not a monomial.
3. [tex]\((x - 2x)^2\)[/tex]:
- Firstly, simplify the expression inside the parentheses: [tex]\(x - 2x = -x\)[/tex]. Then, square the result: [tex]\((-x)^2 = x^2\)[/tex]. The result [tex]\(x^2\)[/tex] is a monomial.
4. [tex]\(\frac{rs}{t}\)[/tex]:
- This expression is not a monomial because it involves a fraction with a variable [tex]\(t\)[/tex] in the denominator.
5. [tex]\(36x^2yz^3\)[/tex]:
- This expression is a product of constants and variables, where each variable has a non-negative integer exponent. Hence, [tex]\(36x^2yz^3\)[/tex] is a monomial.
6. [tex]\(a^x\)[/tex]:
- This expression is not a monomial because the exponent [tex]\(x\)[/tex] is a variable. Monomials must have variable exponents that are non-negative integers (constants).
### Conclusion:
The expressions that are monomials are:
1. [tex]\((-4 + 6)\)[/tex] which simplifies to [tex]\(2\)[/tex].
2. [tex]\((x - 2x)^2\)[/tex] which simplifies to [tex]\(x^2\)[/tex].
3. [tex]\(36x^2yz^3\)[/tex].
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