Answer :

To determine which monomial is a perfect cube, we need to consider the coefficients and whether they are perfect cubes. Here are the steps to identify the perfect cube from a given list of monomials:

1. Understanding a Perfect Cube:
A number is a perfect cube if it can be expressed as [tex]\( n^3 \)[/tex] where [tex]\( n \)[/tex] is an integer. For example, [tex]\( 1^3 = 1 \)[/tex], [tex]\( 8 = 2^3 \)[/tex], etc.

2. Identify the Coefficients:
Look at the coefficients of the given monomials:
- [tex]\( 1 x^3 \)[/tex] has a coefficient of 1.
- [tex]\( 3 x^3 \)[/tex] has a coefficient of 3.
- [tex]\( 6 x^3 \)[/tex] has a coefficient of 6.
- [tex]\( 9 x^3 \)[/tex] has a coefficient of 9.

3. Examine Each Coefficient:
- 1: [tex]\( 1 \)[/tex] can be written as [tex]\( 1^3 \)[/tex] since [tex]\( 1 \times 1 \times 1 = 1 \)[/tex]. Therefore, 1 is a perfect cube.
- 3: [tex]\( 3 \)[/tex] cannot be written as [tex]\( n^3 \)[/tex] for any integer [tex]\( n \)[/tex].
- 6: [tex]\( 6 \)[/tex] cannot be written as [tex]\( n^3 \)[/tex] for any integer [tex]\( n \)[/tex].
- 9: [tex]\( 9 \)[/tex] cannot be written as [tex]\( n^3 \)[/tex] for any integer [tex]\( n \)[/tex]. [tex]\( 9 \)[/tex] can be written as [tex]\( 3^2 \)[/tex] but not as a cube of any integer.

After examining the coefficients, we find that only the coefficient 1 is a perfect cube.

So, the correct monomial that is a perfect cube is [tex]\( 1 x^3 \)[/tex].

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