To determine which expression is a perfect cube, we need to ensure that each component of the expression - the constant and the exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] - fits the criteria for being a perfect cube. Specifically:
1. The constant must be a perfect cube.
2. The exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] need to be multiples of 3.
Let's evaluate each option step-by-step:
### Option 1: [tex]\(12x^{12}y^{18}\)[/tex]
- Constant: 12 is not a perfect cube (as [tex]\(12^{1/3}\)[/tex] is not an integer).
- Exponent of [tex]\(x\)[/tex]: 12 is a multiple of 3.
- Exponent of [tex]\(y\)[/tex]: 18 is a multiple of 3.
Although the exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] meet the criteria, the constant 12 is not a perfect cube. Therefore, this expression is not a perfect cube.
### Option 2: [tex]\(12x^{27}y^{125}\)[/tex]
- Constant: 12 is not a perfect cube.
- Exponent of [tex]\(x\)[/tex]: 27 is a multiple of 3.
- Exponent of [tex]\(y\)[/tex]: 125 is not a multiple of 3.
Here, neither the constant nor the exponent of [tex]\(y\)[/tex] meets the criteria for being a perfect cube. Hence, this expression is not a perfect cube.
### Option 3: [tex]\(64x^{15}y^{18}\)[/tex]
- Constant: 64 is a perfect cube, as [tex]\(64 = 4^3\)[/tex].
- Exponent of [tex]\(x\)[/tex]: 15 is a multiple of 3.
- Exponent of [tex]\(y\)[/tex]: 18 is a multiple of 3.
In this option, the constant is a perfect cube and both the exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are multiples of 3. Therefore, this expression is a perfect cube.
### Option 4: [tex]\(64x^{27}y^{125}\)[/tex]
- Constant: 64 is a perfect cube.
- Exponent of [tex]\(x\)[/tex]: 27 is a multiple of 3.
- Exponent of [tex]\(y\)[/tex]: 125 is not a multiple of 3.
While the constant and the exponent of [tex]\(x\)[/tex] meet the criteria, the exponent of [tex]\(y\)[/tex] does not. Therefore, this expression is not a perfect cube.
After evaluating all options, we conclude that the expression [tex]\(64x^{15}y^{18}\)[/tex] is a perfect cube.
Thus, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
