Answer :
- A perfect cube is a number that is the cube of an integer.
- Check each given volume to see if its cube root is an integer.
- $1 = 1^3$, $8 = 2^3$, $27 = 3^3$, $64 = 4^3$.
- The perfect cubes are $\boxed{1, 8, 27, 64}$ in.$^3$
### Explanation
1. Understanding Perfect Cubes
We are asked to identify which of the given volumes are perfect cubes. A perfect cube is a number that can be obtained by cubing an integer (raising an integer to the power of 3). We need to check each volume to see if its cube root is an integer.
2. Checking Each Volume
Let's check each volume:
* 1: $1 = 1^3$, so 1 is a perfect cube.
* 4: The cube root of 4 is approximately 1.587, which is not an integer, so 4 is not a perfect cube.
* 8: $8 = 2^3$, so 8 is a perfect cube.
* 12: The cube root of 12 is approximately 2.289, which is not an integer, so 12 is not a perfect cube.
* 25: The cube root of 25 is approximately 2.924, which is not an integer, so 25 is not a perfect cube.
* 27: $27 = 3^3$, so 27 is a perfect cube.
* 64: $64 = 4^3$, so 64 is a perfect cube.
3. Identifying Perfect Cubes
Therefore, the volumes that are perfect cubes are 1, 8, 27, and 64.
4. Final Answer
The volumes that represent a perfect cube are $\boxed{1, 8, 27, 64}$.
### Examples
Perfect cubes are useful in various real-world applications. For example, when designing cubic storage containers, knowing perfect cubes helps determine the side lengths needed to achieve specific volumes efficiently. In architecture, understanding perfect cubes aids in creating symmetrical and aesthetically pleasing designs, ensuring structural integrity and visual harmony.
- Check each given volume to see if its cube root is an integer.
- $1 = 1^3$, $8 = 2^3$, $27 = 3^3$, $64 = 4^3$.
- The perfect cubes are $\boxed{1, 8, 27, 64}$ in.$^3$
### Explanation
1. Understanding Perfect Cubes
We are asked to identify which of the given volumes are perfect cubes. A perfect cube is a number that can be obtained by cubing an integer (raising an integer to the power of 3). We need to check each volume to see if its cube root is an integer.
2. Checking Each Volume
Let's check each volume:
* 1: $1 = 1^3$, so 1 is a perfect cube.
* 4: The cube root of 4 is approximately 1.587, which is not an integer, so 4 is not a perfect cube.
* 8: $8 = 2^3$, so 8 is a perfect cube.
* 12: The cube root of 12 is approximately 2.289, which is not an integer, so 12 is not a perfect cube.
* 25: The cube root of 25 is approximately 2.924, which is not an integer, so 25 is not a perfect cube.
* 27: $27 = 3^3$, so 27 is a perfect cube.
* 64: $64 = 4^3$, so 64 is a perfect cube.
3. Identifying Perfect Cubes
Therefore, the volumes that are perfect cubes are 1, 8, 27, and 64.
4. Final Answer
The volumes that represent a perfect cube are $\boxed{1, 8, 27, 64}$.
### Examples
Perfect cubes are useful in various real-world applications. For example, when designing cubic storage containers, knowing perfect cubes helps determine the side lengths needed to achieve specific volumes efficiently. In architecture, understanding perfect cubes aids in creating symmetrical and aesthetically pleasing designs, ensuring structural integrity and visual harmony.
