Answer :

To determine which monomial is a perfect cube, we need to check if the coefficient of the monomial is a perfect cube. A number is a perfect cube if you can multiply an integer by itself three times to get that number.

Let's examine the given monomials:

1. [tex]\(1x^3\)[/tex]: The coefficient is 1.
- The cube root of 1 is 1 ([tex]\(1 \times 1 \times 1 = 1\)[/tex]), which is an integer.
- Therefore, [tex]\(1x^3\)[/tex] is a perfect cube.

2. [tex]\(3x^3\)[/tex]: The coefficient is 3.
- The cube root of 3 is not an integer, since there is no integer that multiplied by itself three times equals 3.
- Therefore, [tex]\(3x^3\)[/tex] is not a perfect cube.

3. [tex]\(6x^3\)[/tex]: The coefficient is 6.
- The cube root of 6 is not an integer, as there is no integer that multiplied by itself three times equals 6.
- Therefore, [tex]\(6x^3\)[/tex] is not a perfect cube.

4. [tex]\(9x^3\)[/tex]: The coefficient is 9.
- The cube root of 9 is not an integer, because there is no integer that multiplied by itself three times equals 9.
- Therefore, [tex]\(9x^3\)[/tex] is not a perfect cube.

After checking each monomial, we find that only [tex]\(1x^3\)[/tex] is a perfect cube.

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