To determine which of the given volumes are perfect cubes, we need to identify numbers that result from cubing an integer. Let’s go through each volume in the list to check if it is a perfect cube:
1. 1 in.³:
- The cube of 1 is [tex]\(1^3 = 1\)[/tex].
- Therefore, 1 in.³ is a perfect cube.
2. 4 in.³:
- We check if there is an integer such that its cube is 4.
- The cube roots of 4 are not integers (approximately 1.587), so 4 in.³ is not a perfect cube.
3. 8 in.³:
- The cube of 2 is [tex]\(2^3 = 8\)[/tex].
- Therefore, 8 in.³ is a perfect cube.
4. 12 in.³:
- We check if there is an integer such that its cube is 12.
- The cube roots of 12 are not integers (approximately 2.289), so 12 in.³ is not a perfect cube.
5. 25 in.³:
- We check if there is an integer such that its cube is 25.
- The cube roots of 25 are not integers (approximately 2.924), so 25 in.³ is not a perfect cube.
6. 27 in.³:
- The cube of 3 is [tex]\(3^3 = 27\)[/tex].
- Therefore, 27 in.³ is a perfect cube.
7. 64 in.³:
- The cube of 4 is [tex]\(4^3 = 64\)[/tex].
- Therefore, 64 in.³ is a perfect cube.
So, the volumes 1 in.³, 8 in.³, 27 in.³, and 64 in.³ are perfect cubes.
