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Answer :
To determine which monomial is a perfect cube, we need to consider each option given and see if the coefficient and the variable together form a perfect cube.
The monomials given are:
1. [tex]\(1 x^3\)[/tex]
2. [tex]\(3 x^3\)[/tex]
3. [tex]\(6 x^3\)[/tex]
4. [tex]\(9 x^3\)[/tex]
A monomial is a perfect cube if both the numerical coefficient and the variable part (here [tex]\(x^3\)[/tex]) are perfect cubes. Since the variable part, [tex]\(x^3\)[/tex], is already a perfect cube ([tex]\(x\)[/tex] is multiplied by itself three times), we just need to check if the numerical coefficients are perfect cubes.
1. [tex]\(1 x^3\)[/tex]:
- The coefficient is 1.
- 1 is a perfect cube because [tex]\(1^3 = 1\)[/tex].
2. [tex]\(3 x^3\)[/tex]:
- The coefficient is 3.
- 3 is not a perfect cube because there is no whole number [tex]\(n\)[/tex] such that [tex]\(n^3 = 3\)[/tex].
3. [tex]\(6 x^3\)[/tex]:
- The coefficient is 6.
- 6 is not a perfect cube because there is no whole number [tex]\(n\)[/tex] such that [tex]\(n^3 = 6\)[/tex].
4. [tex]\(9 x^3\)[/tex]:
- The coefficient is 9.
- 9 is not a perfect cube because there is no whole number [tex]\(n\)[/tex] such that [tex]\(n^3 = 9\)[/tex].
From the analysis, only the monomial [tex]\(1 x^3\)[/tex] is a perfect cube, since its coefficient, 1, is a perfect cube. Therefore, the answer is [tex]\(1 x^3\)[/tex].
The monomials given are:
1. [tex]\(1 x^3\)[/tex]
2. [tex]\(3 x^3\)[/tex]
3. [tex]\(6 x^3\)[/tex]
4. [tex]\(9 x^3\)[/tex]
A monomial is a perfect cube if both the numerical coefficient and the variable part (here [tex]\(x^3\)[/tex]) are perfect cubes. Since the variable part, [tex]\(x^3\)[/tex], is already a perfect cube ([tex]\(x\)[/tex] is multiplied by itself three times), we just need to check if the numerical coefficients are perfect cubes.
1. [tex]\(1 x^3\)[/tex]:
- The coefficient is 1.
- 1 is a perfect cube because [tex]\(1^3 = 1\)[/tex].
2. [tex]\(3 x^3\)[/tex]:
- The coefficient is 3.
- 3 is not a perfect cube because there is no whole number [tex]\(n\)[/tex] such that [tex]\(n^3 = 3\)[/tex].
3. [tex]\(6 x^3\)[/tex]:
- The coefficient is 6.
- 6 is not a perfect cube because there is no whole number [tex]\(n\)[/tex] such that [tex]\(n^3 = 6\)[/tex].
4. [tex]\(9 x^3\)[/tex]:
- The coefficient is 9.
- 9 is not a perfect cube because there is no whole number [tex]\(n\)[/tex] such that [tex]\(n^3 = 9\)[/tex].
From the analysis, only the monomial [tex]\(1 x^3\)[/tex] is a perfect cube, since its coefficient, 1, is a perfect cube. Therefore, the answer is [tex]\(1 x^3\)[/tex].
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