Answer :

To determine which monomial is a perfect cube, let's look at each option and identify if both the coefficient and the variable expression form a perfect cube.

A perfect cube is a number that can be expressed as the cube of an integer. For example, [tex]\(8\)[/tex] is a perfect cube because [tex]\(2^3 = 8\)[/tex].

Now, let's examine each monomial:

1. [tex]\(1x^3\)[/tex]
- The coefficient is [tex]\(1\)[/tex], and [tex]\(1\)[/tex] is a perfect cube since [tex]\(1^3 = 1\)[/tex].
- The variable [tex]\(x^3\)[/tex] is already in the form of a cube ([tex]\(x\)[/tex] raised to the power of [tex]\(3\)[/tex]).
- Therefore, [tex]\(1x^3\)[/tex] is a perfect cube.

2. [tex]\(3x^3\)[/tex]
- The coefficient is [tex]\(3\)[/tex], and [tex]\(3\)[/tex] is not a perfect cube since there is no integer whose cube is [tex]\(3\)[/tex].
- Therefore, [tex]\(3x^3\)[/tex] is not a perfect cube.

3. [tex]\(6x^3\)[/tex]
- The coefficient is [tex]\(6\)[/tex], and [tex]\(6\)[/tex] is not a perfect cube since there is no integer whose cube is [tex]\(6\)[/tex].
- Therefore, [tex]\(6x^3\)[/tex] is not a perfect cube.

4. [tex]\(9x^3\)[/tex]
- The coefficient is [tex]\(9\)[/tex], and [tex]\(9\)[/tex] is not a perfect cube since [tex]\(9^{1/3}\)[/tex] is not an integer.
- Therefore, [tex]\(9x^3\)[/tex] is not a perfect cube.

Based on these evaluations, the monomial [tex]\(1x^3\)[/tex] is the perfect cube.

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