Certainly! Let's analyze each of the given expressions and determine why they are or are not monomials.
### Why the following expressions are not monomials:
1. [tex]\( 3 c d^X \)[/tex]
- Reason: A monomial must have non-variable exponents (i.e., the exponent must be a constant). This expression contains a variable exponent [tex]\( X \)[/tex], which does not meet the criteria of a monomial.
2. [tex]\( x + 2 w \)[/tex]
- Reason: A monomial is a single term. The expression [tex]\( x + 2 w \)[/tex] is a sum of two terms ([tex]\( x \)[/tex] and [tex]\( 2w \)[/tex]), which means it is not a monomial.
3. [tex]\( \frac{3}{h} \)[/tex]
- Reason: A monomial cannot have variables in the denominator. The variable [tex]\( h \)[/tex] is present in the denominator of this fraction, making it not a monomial.
4. [tex]\( a b^{-1} \)[/tex]
- Reason: In a monomial, exponents must be non-negative whole numbers. This expression contains [tex]\( b^{-1} \)[/tex], which represents a negative exponent. Hence, it’s not a monomial.
### Evaluating the given expressions to identify monomials:
1. [tex]\( -4 + 6 \)[/tex]
- Evaluation: This is the sum of two constants ([tex]\(-4\)[/tex] and [tex]\(6\)[/tex]), which totals [tex]\(2\)[/tex]. However, monomials are single terms, not sums of terms, so [tex]\( -4 + 6 \)[/tex] is not a monomial.
2. [tex]\( b + 2 b + 2 \)[/tex]
- Evaluation: This simplifies to [tex]\( 3b + 2 \)[/tex], which is a sum of terms. Since a monomial cannot be a sum of terms, [tex]\( b + 2b + 2 \)[/tex] is not a monomial.
3. [tex]\( (x - 2x)^2 \)[/tex]
- Evaluation: Inside the parentheses, [tex]\( x - 2x = -x \)[/tex], and then squaring that gives [tex]\( (-x)^2 = x^2 \)[/tex]. While it seems it could be a monomial, the initial expression [tex]\( (x - 2x)^2 \)[/tex] simplifies to [tex]\( 0^2 \)[/tex], which is just [tex]\( 0 \)[/tex], not a monomial.
4. [tex]\( \frac{r s}{t} \)[/tex]
- Evaluation: A monomial cannot have variables in the denominator. The variable [tex]\( t \)[/tex] is in the denominator, so [tex]\( \frac{r s}{t} \)[/tex] is not a monomial.
5. [tex]\( 36x^2 y z^3 \)[/tex]
- Evaluation: This is a single term where the exponents of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] are all non-negative integers. Hence, [tex]\( 36x^2 y z^3 \)[/tex] is indeed a monomial.
6. [tex]\( a^x \)[/tex]
- Evaluation: This expression has a variable exponent [tex]\( x \)[/tex]. Since monomials must have constant exponents, [tex]\( a^x \)[/tex] is not a monomial.
7. [tex]\( x^{\frac{1}{3}} \)[/tex]
- Evaluation: The exponent [tex]\(\frac{1}{3}\)[/tex] is a fraction, and exponents in a monomial must be non-negative integers. Hence, this expression is not a monomial.
### Conclusion:
- The valid monomial from the list is:
- [tex]\( 36x^2 y z^3 \)[/tex]
- The expressions [tex]\(3 c d^X\)[/tex], [tex]\( x+2 w\)[/tex], [tex]\( \frac{3}{h} \)[/tex], and [tex]\( a b^{-1} \)[/tex] are not monomials for the reasons explained above.
