To determine which monomials are perfect cubes, we need to check if the coefficients (1, 3, 6, 9) are perfect cubes. A perfect cube is a number that can be expressed as the cube of an integer. Let's examine each coefficient one by one.
1. Checking 1:
- [tex]\(1 = 1^3\)[/tex]
- The integer 1 cubed is indeed 1, so 1 is a perfect cube.
2. Checking 3:
- The cube of an integer closest to 3 would be [tex]\(1^3 = 1\)[/tex] and [tex]\(2^3 = 8\)[/tex]. Since there's no integer value for which the cube is exactly 3, 3 is not a perfect cube.
3. Checking 6:
- The cube of an integer closest to 6 would be [tex]\(1^3 = 1\)[/tex] and [tex]\(2^3 = 8\)[/tex]. Since there's no integer value for which the cube is exactly 6, 6 is not a perfect cube.
4. Checking 9:
- The cube of an integer closest to 9 would be [tex]\(2^3 = 8\)[/tex] and [tex]\(3^3 = 27\)[/tex]. Since there's no integer value for which the cube is exactly 9, 9 is not a perfect cube.
After evaluating each coefficient, we determine that only the number 1 is a perfect cube.
Therefore, the monomial that is a perfect cube is:
[tex]\[ 1 x^3 \][/tex]
