To determine why the given expressions are or are not monomials, let’s analyze each one step-by-step.
### Expression 1: [tex]\( 3xy^2 \)[/tex]
Analysis:
1. A monomial is defined as a single term consisting of a non-zero constant coefficient multiplied by variables raised to non-negative integer exponents.
2. The term [tex]\( 3xy^2 \)[/tex] consists of a coefficient 3 and two variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] where [tex]\( x \)[/tex] is raised to the power of 1 (which is a non-negative integer) and [tex]\( y \)[/tex] is raised to the power of 2 (which is also a non-negative integer).
3. Since there are no additions or subtractions and all exponents are non-negative integers, [tex]\( 3xy^2 \)[/tex] is indeed a monomial.
Conclusion: This expression is a monomial.
### Expression 2: [tex]\( r - 2s \)[/tex]
Analysis:
1. A monomial should be a single term, whereas a polynomial with multiple terms joined by addition or subtraction operations is not a monomial.
2. The expression [tex]\( r - 2s \)[/tex] consists of two terms: [tex]\( r \)[/tex] and [tex]\( -2s \)[/tex].
3. The presence of the subtraction operation makes it a binomial, not a monomial.
Conclusion: This expression is not a monomial because it has two terms.
### Expression 3: [tex]\( (ab)^{\frac{1}{2}} \)[/tex]
Analysis:
1. For an expression to be a monomial, all exponents of the variables must be non-negative integers.
2. The expression [tex]\( (ab)^{\frac{1}{2}} \)[/tex] involves the variables [tex]\( a \)[/tex] and [tex]\( b \)[/tex] raised to the power of [tex]\(\frac{1}{2}\)[/tex], which is a fractional exponent.
3. The presence of a fractional exponent means that [tex]\( (ab)^{\frac{1}{2}} \)[/tex] does not meet the requirement of having non-negative integer exponents.
Conclusion: This expression is not a monomial because it has a fractional exponent.
To summarize:
- [tex]\( 3xy^2 \)[/tex] is a monomial.
- [tex]\( r - 2s \)[/tex] is not a monomial because it has two terms.
- [tex]\( (ab)^{\frac{1}{2}} \)[/tex] is not a monomial because it has a fractional exponent.
