Answer :
To determine which monomial is a perfect cube, we need to focus on the coefficients of each monomial and identify if any of them are perfect cubes.
Here's a step-by-step approach to solving this problem:
1. Identify the monomials and their coefficients:
- Monomial 1: [tex]\(1x^3\)[/tex] has the coefficient 1.
- Monomial 2: [tex]\(3x^3\)[/tex] has the coefficient 3.
- Monomial 3: [tex]\(6x^3\)[/tex] has the coefficient 6.
- Monomial 4: [tex]\(9x^3\)[/tex] has the coefficient 9.
2. Understand what a perfect cube is:
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], [tex]\(8\)[/tex] (since [tex]\(2^3 = 8\)[/tex]), and [tex]\(27\)[/tex] (since [tex]\(3^3 = 27\)[/tex]) are perfect cubes.
3. Check each coefficient to see if it is a perfect cube:
- For the coefficient 1: It is a perfect cube because [tex]\(1^3 = 1\)[/tex].
- For the coefficient 3: It is not a perfect cube because there is no integer [tex]\(n\)[/tex] such that [tex]\(n^3 = 3\)[/tex].
- For the coefficient 6: It is not a perfect cube because there is no integer [tex]\(n\)[/tex] such that [tex]\(n^3 = 6\)[/tex].
- For the coefficient 9: It is not a perfect cube because there is no integer [tex]\(n\)[/tex] such that [tex]\(n^3 = 9\)[/tex].
4. Conclusion:
The only coefficient that is a perfect cube is 1. Therefore, the monomial [tex]\(1x^3\)[/tex] is a perfect cube.
Here's a step-by-step approach to solving this problem:
1. Identify the monomials and their coefficients:
- Monomial 1: [tex]\(1x^3\)[/tex] has the coefficient 1.
- Monomial 2: [tex]\(3x^3\)[/tex] has the coefficient 3.
- Monomial 3: [tex]\(6x^3\)[/tex] has the coefficient 6.
- Monomial 4: [tex]\(9x^3\)[/tex] has the coefficient 9.
2. Understand what a perfect cube is:
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], [tex]\(8\)[/tex] (since [tex]\(2^3 = 8\)[/tex]), and [tex]\(27\)[/tex] (since [tex]\(3^3 = 27\)[/tex]) are perfect cubes.
3. Check each coefficient to see if it is a perfect cube:
- For the coefficient 1: It is a perfect cube because [tex]\(1^3 = 1\)[/tex].
- For the coefficient 3: It is not a perfect cube because there is no integer [tex]\(n\)[/tex] such that [tex]\(n^3 = 3\)[/tex].
- For the coefficient 6: It is not a perfect cube because there is no integer [tex]\(n\)[/tex] such that [tex]\(n^3 = 6\)[/tex].
- For the coefficient 9: It is not a perfect cube because there is no integer [tex]\(n\)[/tex] such that [tex]\(n^3 = 9\)[/tex].
4. Conclusion:
The only coefficient that is a perfect cube is 1. Therefore, the monomial [tex]\(1x^3\)[/tex] is a perfect cube.
