Answer :
To determine which monomial is a perfect cube from the options given, we need to analyze the coefficient of each monomial because the variable part, [tex]\(x^3\)[/tex], is already a cube.
Monomials involved are:
1. [tex]\(1x^3\)[/tex]
2. [tex]\(3x^3\)[/tex]
3. [tex]\(6x^3\)[/tex]
4. [tex]\(9x^3\)[/tex]
Let's look at each coefficient to see if it is a perfect cube. A number is a perfect cube if it can be written as some integer raised to the power of three.
1. Coefficient: 1
- [tex]\(1 = 1^3\)[/tex], which means 1 is a perfect cube.
2. Coefficient: 3
- There is no integer [tex]\(n\)[/tex] such that [tex]\(n^3 = 3\)[/tex]. Therefore, 3 is not a perfect cube.
3. Coefficient: 6
- There is no integer [tex]\(n\)[/tex] such that [tex]\(n^3 = 6\)[/tex]. Therefore, 6 is not a perfect cube.
4. Coefficient: 9
- There is no integer [tex]\(n\)[/tex] such that [tex]\(n^3 = 9\)[/tex]. Therefore, 9 is not a perfect cube.
Based on this examination, the only monomial among the options with a coefficient that is a perfect cube is [tex]\(1x^3\)[/tex]. Therefore, [tex]\(1x^3\)[/tex] is the monomial that is a perfect cube.
Monomials involved are:
1. [tex]\(1x^3\)[/tex]
2. [tex]\(3x^3\)[/tex]
3. [tex]\(6x^3\)[/tex]
4. [tex]\(9x^3\)[/tex]
Let's look at each coefficient to see if it is a perfect cube. A number is a perfect cube if it can be written as some integer raised to the power of three.
1. Coefficient: 1
- [tex]\(1 = 1^3\)[/tex], which means 1 is a perfect cube.
2. Coefficient: 3
- There is no integer [tex]\(n\)[/tex] such that [tex]\(n^3 = 3\)[/tex]. Therefore, 3 is not a perfect cube.
3. Coefficient: 6
- There is no integer [tex]\(n\)[/tex] such that [tex]\(n^3 = 6\)[/tex]. Therefore, 6 is not a perfect cube.
4. Coefficient: 9
- There is no integer [tex]\(n\)[/tex] such that [tex]\(n^3 = 9\)[/tex]. Therefore, 9 is not a perfect cube.
Based on this examination, the only monomial among the options with a coefficient that is a perfect cube is [tex]\(1x^3\)[/tex]. Therefore, [tex]\(1x^3\)[/tex] is the monomial that is a perfect cube.
