To determine which numbers from the given set are perfect cubes, we need to assess each one individually. Here's the step-by-step process:
1. Evaluate 64:
- A perfect cube is a number that can be written as [tex]\( n^3 \)[/tex], where [tex]\( n \)[/tex] is an integer.
- To test if 64 is a perfect cube, find the cube root of 64.
- The cube root of 64 is [tex]\( \sqrt[3]{64} = 4 \)[/tex], since [tex]\( 4^3 = 64 \)[/tex].
- Therefore, 64 is a perfect cube.
2. Evaluate [tex]\( x^{16} \)[/tex]:
- Consider whether [tex]\( x^{16} \)[/tex] can be written in the form of [tex]\( (x^k)^3 \)[/tex].
- [tex]\( x^{16} \)[/tex] would need to be of the form [tex]\( x^{3k} \)[/tex]. However, since 16 is not divisible by 3, [tex]\( x^{16} \)[/tex] cannot be expressed as [tex]\( (x^k)^3 \)[/tex].
- Therefore, [tex]\( x^{16} \)[/tex] is not a perfect cube.
3. Evaluate [tex]\( 8 x^3 \)[/tex]:
- A perfect cube can be written in the form [tex]\( (n m)^3 \)[/tex].
- Here, we have [tex]\( 8 \)[/tex], which is [tex]\( 2^3 \)[/tex], and [tex]\( x^3 \)[/tex], which is also a perfect cube.
- Therefore, [tex]\( 8 x^3 \)[/tex] is a perfect cube.
4. Evaluate [tex]\( 27 x^4 \)[/tex]:
- [tex]\( 27 \)[/tex] is a perfect cube because [tex]\( 27 = 3^3 \)[/tex].
- However, [tex]\( x^4 \)[/tex] is not a perfect cube because 4 is not divisible by 3.
- Therefore, [tex]\( 27 x^4 \)[/tex] is not a perfect cube.
5. Evaluate [tex]\( 81 x^6 \)[/tex]:
- [tex]\( 81 \)[/tex] is not a perfect cube because [tex]\( 81 = 3^4 \)[/tex], and 3 is not a perfect cube itself.
- [tex]\( x^6 \)[/tex] is a perfect cube since [tex]\( x^6 = (x^2)^3 \)[/tex].
- As [tex]\( 81 \)[/tex] is not a perfect cube, [tex]\( 81 x^6 \)[/tex] itself isn't a perfect cube.
- Therefore, [tex]\( 81 x^6 \)[/tex] is not a perfect cube.
6. Evaluate [tex]\( 125 x^9 \)[/tex]:
- [tex]\( 125 \)[/tex] is a perfect cube because [tex]\( 125 = 5^3 \)[/tex].
- [tex]\( x^9 \)[/tex] is also a perfect cube because [tex]\( x^9 = (x^3)^3 \)[/tex].
- Hence, [tex]\( 125 x^9 \)[/tex] is a perfect cube.
After evaluating each option, the results indicate:
- [tex]\( 64 \)[/tex]
- [tex]\( 8 x^3 \)[/tex]
- [tex]\( 125 x^9 \)[/tex]
These are the terms that are perfect cubes. Among these, considering only numerical values:
64 is confirmed as a perfect cube in the set given.
