To solve the question of identifying which expressions are monomials, let's understand what a monomial is. A monomial is an algebraic expression that is a single term consisting of a constant and/or a variable raised to a non-negative integer exponent. Monomials do not involve addition, subtraction, or variables in denominators or under radicals.
Let's analyze each option given:
A. [tex]\( x \)[/tex]
This is a single variable with the exponent 1 (which is a non-negative integer). Therefore, [tex]\( x \)[/tex] is a monomial.
B. [tex]\( \sqrt{x} \)[/tex]
The square root of [tex]\( x \)[/tex] can be written as [tex]\( x^{1/2} \)[/tex]. Since the exponent is not an integer, [tex]\( \sqrt{x} \)[/tex] is not a monomial.
C. 6
This is a constant term, which can be considered as [tex]\( 6 \cdot x^0 \)[/tex] (where the exponent is 0, a non-negative integer). Therefore, 6 is a monomial.
D. [tex]\( x + 1 \)[/tex]
This expression involves the addition of two terms. Monomials do not involve addition or subtraction, hence [tex]\( x + 1 \)[/tex] is not a monomial.
E. [tex]\( -4 x^3 \)[/tex]
This is a single term where the variable [tex]\( x \)[/tex] is raised to the power of 3 (a non-negative integer) and multiplied by a constant. Hence, [tex]\( -4 x^3 \)[/tex] is a monomial.
F. [tex]\( \frac{5}{x} \)[/tex]
This can be written as [tex]\( 5x^{-1} \)[/tex]. Since the exponent is negative, [tex]\( \frac{5}{x} \)[/tex] is not considered a monomial.
Based on this analysis, the expressions that are monomials are:
- A. [tex]\( x \)[/tex]
- C. 6
- E. [tex]\( -4 x^3 \)[/tex]
Therefore, the monomials can be listed as:
```
['A', 'C', 'E']
```
