Answer :
To determine which monomial is a perfect cube, let's focus on the coefficients provided with the term [tex]\(x^3\)[/tex]. The monomials we need to analyze are:
- [tex]\(1 \cdot x^3\)[/tex]
- [tex]\(3 \cdot x^3\)[/tex]
- [tex]\(6 \cdot x^3\)[/tex]
- [tex]\(9 \cdot x^3\)[/tex]
Since all the terms share the same [tex]\(x^3\)[/tex] factor, we only need to consider if the coefficients themselves are perfect cubes.
Definition:
A perfect cube is a number that can be written as [tex]\(n^3\)[/tex], where [tex]\(n\)[/tex] is an integer. For example, [tex]\(1\)[/tex] is a perfect cube because it can be written as [tex]\(1^3\)[/tex], which equals [tex]\(1\)[/tex].
Steps:
1. Check each coefficient:
- 1: [tex]\(1^3 = 1\)[/tex]. Thus, 1 is a perfect cube.
- 3: [tex]\(3\)[/tex] is not a perfect cube since no integer [tex]\(n\)[/tex] satisfies [tex]\(n^3 = 3\)[/tex].
- 6: [tex]\(6\)[/tex] is not a perfect cube since there's no integer whose cube is 6.
- 9: [tex]\(9\)[/tex] is not a perfect cube because [tex]\(2^3 = 8\)[/tex] and [tex]\(3^3 = 27\)[/tex], and neither of these is 9.
2. Conclusion:
The only coefficient in the given monomials that is a perfect cube is 1. Therefore, the monomial [tex]\(1 \cdot x^3 = x^3\)[/tex] is a perfect cube.
So, the monomial [tex]\(x^3\)[/tex] (with the coefficient 1) is a perfect cube.
- [tex]\(1 \cdot x^3\)[/tex]
- [tex]\(3 \cdot x^3\)[/tex]
- [tex]\(6 \cdot x^3\)[/tex]
- [tex]\(9 \cdot x^3\)[/tex]
Since all the terms share the same [tex]\(x^3\)[/tex] factor, we only need to consider if the coefficients themselves are perfect cubes.
Definition:
A perfect cube is a number that can be written as [tex]\(n^3\)[/tex], where [tex]\(n\)[/tex] is an integer. For example, [tex]\(1\)[/tex] is a perfect cube because it can be written as [tex]\(1^3\)[/tex], which equals [tex]\(1\)[/tex].
Steps:
1. Check each coefficient:
- 1: [tex]\(1^3 = 1\)[/tex]. Thus, 1 is a perfect cube.
- 3: [tex]\(3\)[/tex] is not a perfect cube since no integer [tex]\(n\)[/tex] satisfies [tex]\(n^3 = 3\)[/tex].
- 6: [tex]\(6\)[/tex] is not a perfect cube since there's no integer whose cube is 6.
- 9: [tex]\(9\)[/tex] is not a perfect cube because [tex]\(2^3 = 8\)[/tex] and [tex]\(3^3 = 27\)[/tex], and neither of these is 9.
2. Conclusion:
The only coefficient in the given monomials that is a perfect cube is 1. Therefore, the monomial [tex]\(1 \cdot x^3 = x^3\)[/tex] is a perfect cube.
So, the monomial [tex]\(x^3\)[/tex] (with the coefficient 1) is a perfect cube.
