Answer :
To determine which monomial is a perfect cube, let's look at the options we have: [tex]\(1x^3\)[/tex], [tex]\(3x^3\)[/tex], [tex]\(6x^3\)[/tex], and [tex]\(9x^3\)[/tex].
A monomial is a perfect cube when both the coefficient and the variable part can be written as something cubed. Since all of the options have [tex]\(x^3\)[/tex], we only need to focus on whether the coefficients (1, 3, 6, and 9) are perfect cubes.
1. Check each coefficient:
- Coefficient 1:
The number 1 can be written as [tex]\(1^3 = 1\)[/tex]. So, 1 is a perfect cube.
- Coefficient 3:
Check if there is a whole number that, when cubed, equals 3. There isn't one (as the cube of 1 is 1, and the cube of 2 is 8), so 3 is not a perfect cube.
- Coefficient 6:
Similarly, check for 6. The cube of 1 is 1, and the cube of 2 is 8, so 6 is not a perfect cube.
- Coefficient 9:
Check for 9. The cube of 1 is 1, and the cube of 2 is 8, and the cube of 3 is 27, so 9 is not a perfect cube.
2. Determine which monomial is a perfect cube:
Based on these checks, the only coefficient that is a perfect cube is 1. Therefore, the monomial [tex]\(1x^3\)[/tex] is a perfect cube.
So, the monomial that is a perfect cube is [tex]\(1x^3\)[/tex].
A monomial is a perfect cube when both the coefficient and the variable part can be written as something cubed. Since all of the options have [tex]\(x^3\)[/tex], we only need to focus on whether the coefficients (1, 3, 6, and 9) are perfect cubes.
1. Check each coefficient:
- Coefficient 1:
The number 1 can be written as [tex]\(1^3 = 1\)[/tex]. So, 1 is a perfect cube.
- Coefficient 3:
Check if there is a whole number that, when cubed, equals 3. There isn't one (as the cube of 1 is 1, and the cube of 2 is 8), so 3 is not a perfect cube.
- Coefficient 6:
Similarly, check for 6. The cube of 1 is 1, and the cube of 2 is 8, so 6 is not a perfect cube.
- Coefficient 9:
Check for 9. The cube of 1 is 1, and the cube of 2 is 8, and the cube of 3 is 27, so 9 is not a perfect cube.
2. Determine which monomial is a perfect cube:
Based on these checks, the only coefficient that is a perfect cube is 1. Therefore, the monomial [tex]\(1x^3\)[/tex] is a perfect cube.
So, the monomial that is a perfect cube is [tex]\(1x^3\)[/tex].
