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Answer :
To determine which monomial is a perfect cube, we need to check the coefficients and the variables in each monomial. A perfect cube is a number or expression that can be expressed as something raised to the power of three.
Let's analyze each option one by one:
1. [tex]$1 x^3$[/tex]:
- The coefficient is 1, and [tex]\(1^3 = 1\)[/tex], so 1 is a perfect cube.
- The variable part is [tex]\(x^3\)[/tex], which is already in the form of a cube.
- Therefore, [tex]\(1 x^3\)[/tex] is a perfect cube.
2. [tex]$3 x^3$[/tex]:
- The coefficient is 3. There is no integer [tex]\(a\)[/tex] such that [tex]\(a^3 = 3\)[/tex], so 3 is not a perfect cube.
- The variable part is [tex]\(x^3\)[/tex], which is a cube, but because the coefficient is not a cube, [tex]\(3 x^3\)[/tex] is not a perfect cube.
3. [tex]$6 x^3$[/tex]:
- The coefficient is 6. There is no integer [tex]\(a\)[/tex] such that [tex]\(a^3 = 6\)[/tex], so 6 is not a perfect cube.
- The variable part is [tex]\(x^3\)[/tex], which is a cube, but since the coefficient is not a cube, [tex]\(6 x^3\)[/tex] is not a perfect cube.
4. [tex]$9 x^3$[/tex]:
- The coefficient is 9. Again, there is no integer [tex]\(a\)[/tex] such that [tex]\(a^3 = 9\)[/tex], so 9 is not a perfect cube.
- The variable part is [tex]\(x^3\)[/tex], but because the coefficient is not a cube, [tex]\(9 x^3\)[/tex] is not a perfect cube.
After analyzing each monomial, we conclude that the only monomial that is a perfect cube is [tex]$1 x^3$[/tex].
Let's analyze each option one by one:
1. [tex]$1 x^3$[/tex]:
- The coefficient is 1, and [tex]\(1^3 = 1\)[/tex], so 1 is a perfect cube.
- The variable part is [tex]\(x^3\)[/tex], which is already in the form of a cube.
- Therefore, [tex]\(1 x^3\)[/tex] is a perfect cube.
2. [tex]$3 x^3$[/tex]:
- The coefficient is 3. There is no integer [tex]\(a\)[/tex] such that [tex]\(a^3 = 3\)[/tex], so 3 is not a perfect cube.
- The variable part is [tex]\(x^3\)[/tex], which is a cube, but because the coefficient is not a cube, [tex]\(3 x^3\)[/tex] is not a perfect cube.
3. [tex]$6 x^3$[/tex]:
- The coefficient is 6. There is no integer [tex]\(a\)[/tex] such that [tex]\(a^3 = 6\)[/tex], so 6 is not a perfect cube.
- The variable part is [tex]\(x^3\)[/tex], which is a cube, but since the coefficient is not a cube, [tex]\(6 x^3\)[/tex] is not a perfect cube.
4. [tex]$9 x^3$[/tex]:
- The coefficient is 9. Again, there is no integer [tex]\(a\)[/tex] such that [tex]\(a^3 = 9\)[/tex], so 9 is not a perfect cube.
- The variable part is [tex]\(x^3\)[/tex], but because the coefficient is not a cube, [tex]\(9 x^3\)[/tex] is not a perfect cube.
After analyzing each monomial, we conclude that the only monomial that is a perfect cube is [tex]$1 x^3$[/tex].
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