Answer :
To determine which monomial is a perfect cube, we need to consider both the coefficient (the number in front of the variable) and the exponent of the variable. A perfect cube is a number that can be expressed as the cube of an integer.
Let's analyze each option:
1. [tex]\(1x^3\)[/tex]:
- Coefficient: 1 is a perfect cube because [tex]\(1 = 1^3\)[/tex].
- Variable's exponent: [tex]\(x^3\)[/tex] is a perfect cube form since the exponent is 3.
- Conclusion: [tex]\(1x^3\)[/tex] is a perfect cube.
2. [tex]\(3x^3\)[/tex]:
- Coefficient: 3 is not a perfect cube because there is no integer that multiplies by itself three times to equal 3.
- Conclusion: [tex]\(3x^3\)[/tex] is not a perfect cube.
3. [tex]\(6x^3\)[/tex]:
- Coefficient: 6 is not a perfect cube because there is no integer that multiplies by itself three times to equal 6.
- Conclusion: [tex]\(6x^3\)[/tex] is not a perfect cube.
4. [tex]\(9x^3\)[/tex]:
- Coefficient: 9 is not a perfect cube because there is no integer that multiplies by itself three times to equal 9.
- Conclusion: [tex]\(9x^3\)[/tex] is not a perfect cube.
After evaluating each option, the only monomial that meets the criteria for a perfect cube is [tex]\(1x^3\)[/tex].
Let's analyze each option:
1. [tex]\(1x^3\)[/tex]:
- Coefficient: 1 is a perfect cube because [tex]\(1 = 1^3\)[/tex].
- Variable's exponent: [tex]\(x^3\)[/tex] is a perfect cube form since the exponent is 3.
- Conclusion: [tex]\(1x^3\)[/tex] is a perfect cube.
2. [tex]\(3x^3\)[/tex]:
- Coefficient: 3 is not a perfect cube because there is no integer that multiplies by itself three times to equal 3.
- Conclusion: [tex]\(3x^3\)[/tex] is not a perfect cube.
3. [tex]\(6x^3\)[/tex]:
- Coefficient: 6 is not a perfect cube because there is no integer that multiplies by itself three times to equal 6.
- Conclusion: [tex]\(6x^3\)[/tex] is not a perfect cube.
4. [tex]\(9x^3\)[/tex]:
- Coefficient: 9 is not a perfect cube because there is no integer that multiplies by itself three times to equal 9.
- Conclusion: [tex]\(9x^3\)[/tex] is not a perfect cube.
After evaluating each option, the only monomial that meets the criteria for a perfect cube is [tex]\(1x^3\)[/tex].
